This is just to add a little to luchonacho's answer.
It first describes the legal position of the 39 Articles which is that they are definitive doctrine.
It then looks at the level of Commitment to the Articles required from clergy, describing how this is now much weaker than once it was.
It next considers the breadth in which they can be interpreted and the lack of precision which it is at least possible to hold they contain.
It then suggests that the Articles have been largely neglected and have little practical role in parish life,
Then it looks at the Homilies.
Finally a Summary is given. TL;DR please proceed to summary.
Legal Position of the Thirty-Nine Articles
The legal position is stated in Canon A5 as follows:
The doctrine of the Church of England is grounded in the Holy Scriptures, and in such teachings of the ancient Fathers and Councils of the Church as are agreeable to the said Scriptures.
In particular such doctrine is to be found in the Thirty-nine Articles of Religion, The Book of Common Prayer, and the Ordinal.
and Canon A2 says
The Thirty-nine Articles are agreeable to the Word of God and may be assented unto with a good conscience by all members of the Church of England.
Ref: Canons of the Church of England
The Doctrine and Worship Measure, approved by the UK Parliament in 1974, allows the General Synod to authorise alternative services to those in the Book of Common Prayer provided that
(they) shall be such as in the opinion of the General Synod is neither contrary to, nor indicative of any departure from, the doctrine of the Church of England in any essential matter,
This it may be noted does permit the alternative services to "depart from" the doctrine of the Church provided that the matter is not considered essential.
Degree of Commitment required from Clergy
Ministers in the Church of England (deacons, priests or bishops) are required to affirm their commitment to the Thirty-Nine Articles on ordination. However the way in which this is done has become gradually more woolly. Prior to 1865 it ran
I A B do willingly and from my heart subscribe to the Thirty-nine Articles of Religion of the United Church of England and Ireland ...
(This had been the same since 1689 except the reference to Ireland was introduced only in 1801, when the Act of Union of the Parliaments of Great Britain and Ireland also united the Church of Ireland with the Church of England).
From 1865 to 1975 it was
I A B do solemnly make the following declaration:
I assent to the Thirty-Nine Articles of Religion...
I believe the doctrine of the United Church of England and Ireland, as therein set forth, to be agreeable to the Word of God...
(The reference to Ireland was removed in 1871 when the United Church split into C of E and C of I again.)
Since 1975 a new version has left many people attending ordination services bemused as to what exactly has been affirmed. The comparison with the earlier versions is, I think, important as the purpose of the more wordy statements can be assessed only by comparison with clearer alternatives.
Whoever wrote it was not a believer in "Let your communication be yea or nay". (Matthew 5 37).
The administrator first reads a preface, purporting to interpret the statement the ordinand will make:
The Church of England is part of the One, Holy, Catholic and Apostolic Church worshipping the one true God, Father, Son and Holy Spirit. It professes the faith uniquely revealed in the Holy Scriptures and set forth in the catholic creeds, which faith the Church is called upon to proclaim afresh in each generation. Led by the Holy Spirit, it has borne witness to Christian truth in its historic formularies, the Thirty-nine Articles of Religion, The Book of Common Prayer and the Ordering of Bishops, Priests and Deacons. In the declaration you are about to make will you affirm your loyalty to this inheritance of faith as your inspiration and guidance under God in bringing the grace and truth of Christ to this generation and making him known to those in your care?
According to this, then, the ordinand is about to "affirm his loyalty" to this inheritance of faith. But what the ordinand then says is:
I, A B, do so affirm, and accordingly declare my belief in the faith which is revealed in the Holy Scriptures and set forth in the catholic creeds and to which the historic formularies of the Church of England bear witness
So here he actually does declare his belief in the faith to which the 39 Articles bear witness. Whether this is equivalent to saying he believes the Articles is not very plain.
Ref:Subscription to the Articles
The Meaning and Interpretation of the Thirty-Nine Articles
A great deal of ink has been used, particularly in the Nineteenth Century, in interpreting the Articles in all sorts of ways which were not, to most people, the obvious interpretation of them. Perhaps the most famous was written by the Reverend Newman who, partly prompted by opposition in the Church of England, to his interpretations, became Roman Catholic and eventually became Cardinal Newman. He argued, to the astonishment of most English people, and the Vatican, that the Thirty-Nine Articles were, in fact, completely compatible with the Council of Trent.
Here are a few examples:
Article 31 OF the One Oblation of Christ finished upon the Cross has the words "Wherefore the sacrifices of Masses, in which it was commonly said, the Priest did offer Christ for the quick and the dead, to have remission of pain or guilt, were blasphemous fables and dangerous deceits".
The traditional understanding of this is that the concept of the Roman Catholic Mass as a Sacrifice is being described as a blasphemous fable and a dangerous deceit, which comes across as a distinctly negative assessment. Reverend Newman argued that the phrase "sacrifices of Masses" ought not to be confused with ""the Sacrifice of the Mass" and the article was, in fact, not directed at certain practices occasionally carried out during Mass, but never condoned by Rome.
Article 22 "Of Purgatory" is succinct
The Romish Doctrine concerning Purgatory, Pardons, Worshipping and Adoration, as well of Images as of Relics, and also invocation of Saints, is a fond thing vainly invented, and grounded upon no warranty of Scripture, but rather repugnant to the Word of God.
The traditional Protestant interpretation within the Church of England is that the Church of England does not believe in Purgatory or in praying to saints. But no, said Reverend Newman (as he then was) - it is not Purgatory that the Church of England rejects, it is the Romish doctrine of Purgatory. And the Romish doctrine of Purgatory was not the doctrine affirmed at the Council of Trent (which had not concluded when the Articles were completed), but a previous doctrine once unofficially held in Rome. About the Tridentine Doctrine of Purgatory, about the existence of purgatory and the invocation of saints in general, the Articles are silent.
The final sentence of Article 28 says:
The Sacrament of the Lord's Supper was not by Christ's ordinance reserved, carried about, lifted up or worshipped.
So Reservation of the Sacrament and Corpus Christi processions are a no-no in the C of E; or so it was thought.. But no, just because Christ did not command something does not mean He forbade it. As a matter of simple fact Reservation is not commanded by Christ's ordinance. That doesn't mean it is necessarily a bad thing, or even not an excellent thing.
It is not only from the Catholic side that new interpretations are made. Article 27 "Of Baptism" includes
it is also a sign of Regeneration, or new birth, whereby, as by an instrument, they that receive baptism rightly are grafted into the Church ..
So baptism works as an instrument, which is closer to the Catholic doctrine, except it says "they that receive baptism rightly" and gives no definition as to what distinguishes wrongly from rightly. So perhaps it can be interpreted differently from what at first sight it seems to mean.
A more recent example concerns same-sex marriage. This is now legal in England, but the Church of England does not approve of clergy getting married to persons of the same sex. Those in favour of same-sex marriage for vicars point to Article 32, which was originally written against priests having to remain celibate. It says of ministers that
it is lawful for them , as for all other Christian men, to marry at their own discretion
On this basis the Church, it is argued, ought not to forbid clergy to get married to persons of the same sex.
With the exception of the last example, it is actually not surprising that a wide range of interpretations can be put on the Thirty-Nine Articles. This is because they were designed in the first place to achieve as large a consensus as reasonably possible. There has been much discussion about whether Article 17 is Calvinist or Arminian, for example. It is said the Articles were originally drawn so as not to be problematical to Catholic-minded, as well as Protestant-minded people, and in affixing the interpretations he did, Newman was merely taking advantage of the loopholes deliberately left in to placate people of similar views to his, at the time the Articles were written.
See Tract Ninety for Cardinal Newman's analysis.
From all this, it is, I hope, clear that the Thirty-Nine Articles cannot be very clearly used for teaching purposes. Nor, for most purposes would a reference to them be cited as proof of a doctrinal position. Almost invariable any such proof would come in the form of a citation from Holy Scripture.
General Neglect of the Articles
In 1877 JC Ryle, who was, in 1880, appointed first Protestant Bishop of Liverpool, wrote of the Thirty-Nine Articles that:
It is a melancholy fact, explain it as we may, that for the last 200 years the Articles have fallen into great and undeserved neglect
So Bishop Ryle laments that the Articles had already been largely neglected since the late seventeenth century. There is little evidence to say that they have been less neglected in the period since,. Indeed there are stories of vicars claiming never to have read them, and they do not, in general, feature prominently, if at all, in training at theological colleges.
At the time the homilies were produced many priests were not licenced to preach, due largely to ignorance and lack of education. The opposite situation now prevails - there are many people licenced to preach who are not ordained ministers, principally lay readers. The reading of prepared homilies is not common, at least not overtly. it being generally expected a preacher write his or her own sermon.
Also, the doctrines in the Homilies are described by the Article as godly, wholesome and necessary, but this is short of giving them the same status, in every particular, as the Articles themselves.
Can they be reliably quoted as that which the Church of England now practices and preaches.
Coming to the final part of the question, I can only offer my own observations. It is that nothing, absolutely nothing, can now be said in general. as to what the clergy of the Church of England now preach or practice. There are some who admit to not believing the Resurrection, even Bishop Jenkins of Durham in the 1980s. Some are atheist, and they too claim to believe the Articles in what they say about God's characteristics etc. In the same way they believe that Sherlock Holmes lived in Baker Street, smoked a pipe and had a friend named Watson. They also believe that, like Sherlock Holmes, God has no objective reality. Just about every imaginable views are expressed by clergymen and women. It is said the Church is degenerating into squabbling factions who barely recognise each other as Christians. Some say that, as a national Church, it should contain a comprehensive range of views, but others say that a citizen has the right to expect to be taught the same doctrine in whichever parish he or she lives.
The 39 Articles remain the definitive doctrine of the Church of England.
The clergy are required to subscribe to them, but the form of this subscription is much weakened from earlier years.
The Articles are not entirely clear and are open to a very wide range of interpretations.
They are not, and never have been, prominent in the life of a parish. In practice doctrines, if supported at all, are supported directly from the Bible.
They are sometimes referred to in controversies.
The homilies have fallen into disuse, were never of the same standing as the Articles, but are occasionally quoted in controversies.
The Church of England is now so diverse that neither the Articles nor anything else can truly be said to be representative, in practice, of the Church in general.