While this is easy enough to answer from a Reformed point of view, I'd like to start by pointing out that the felt need for extra-Biblical statements on matters of faith is not limited to Reformed circles or even Protestantism. In fact they are common to all traditions and sects in Christendom.1 Even your most run-of-the-mill non-denominational half-baked-evangelical Church in Podunk middle-America will have a Statement of Faith somewhere on their Bulletin, website or other literature. These statements serve both a declarative role in announcing what they believe are the key points to the Christian faith and a discerning role in separating them from any group or church that would make a contrary statement.
At the core is the issue of language. Words by themselves are of little consequence until they are interpreted. Lets try an example. Let's say I were to make a simple statement as follows:
I do not deserve to be mistreated by the world! -- Caleb Maclennan
You read that and now you think you know what I meant.
Maybe you nod your head in approval—or maybe you wonder what happened to me that I consider myself so mistreated. Maybe it moves you and you start an activist group. You use my statement as a slogan and you gather a bunch of followers. They all are motivated to end the wrongs done by humanity to itself. The only problem is, you completely misunderstood what I said. Now my name is on a whole bunch of things I do not actually approve of.
My statement requires interpretation. Even when you think you are "just reading it at face value", you are actually interpreting it based on whatever background knowledge you have of the world and myself and of language. You also bring your presuppositions about what is right and wrong, what human rights are, etc.
You see, if you really knew my perspective on those factors, a different meaning will show up in the same words. I believe that I am a wretched sinner who does not deserve the grace I have been shown in Christ to whom I owe all of my allegiance. As the author and perfecter of my faith was abused and mistreated by this world having done no wrong to deserve it, I having done great wrong feel unworthy of the lesser mistreatment that I receive. Behind my statement lies an understanding that even the world's mistreatment is better than I deserve.
The need for Creeds and Confessions stems from this issue of all language requiring interpretation. Over time as contexts change and we lose perspective on the original authors, sometimes it becomes necessary to clarify to ourselves in what way we are interpreting the teachings of Scripture.
In reading the Scriptures we each bring different light to bear on them. It is only by comparing our understanding with each others' that we come to realize places where we may have erred, and it is partly by seeing the different outcomes that we come to understand how different presuppositions affect our understanding. Creeds and Confessions help bring interpretation full circle, demonstrating to others how we have interpreted the Scriptures and affirming our beliefs. They are a way of echoing back what we have gleaned from the Bible with the Spirit's help, allowing others to verify our interpretations and hold us accountable to them.
They also serve as a measuring stick for other believers. As a minister of the word ordained by a Reformed denomination, part of the ordination process I went through was an examination of my own beliefs in comparison to a number of Creeds and Confessions, including the Westminster Confession of Faith. I will use the WCF as an example because you mention it, but the principles I'm talking about here and the necessity of such confessions is universal and just as relevant for those who hold a confession that conflicts with the WCF.
I actually took exception to a number of points in the WCF. I do not hold it to be an infallible guide or standard of interpretation nor a dictate on my life. However it is incredibly valuable to able to compare my own interpretation of Scripture (particularly in a case like mine where I go on to teach and preach based on said interpretations) against a third party articulation of the Christian faith. Many people use their Bibles to preach sermons, but just because they used a verse or two doesn't mean they are preaching orthodoxy. In fact many of them babble nonsense and even heresy. Having agreed upon fixed standards of interpretation for certain key issues allows other churches, my own congregation and myself to judge my own beliefs and determine if I have wandered from sound teaching. I do not agree with everything in the WCF, but I have made my exceptions known and that gives people a measure with which to evaluate me and keep my teaching accountable.
Confessions are not divinely appointed standards, but they are a mutual aide for us as fallible individuals that we are correctly interpreting God's word.
The necessity for these started very early in Christian history. It actually can be traced back before the NT church into creedal statements made that existed as markers between Israel and other nations and in between different sects of Jews. Some of these were as simple as a fragment of a verse from the OT, some as complicated as a law dictionary.
But let's pick up some traces from the NT. Please note there are huge gaps between each of these steps, I'm not noting some mile markers along the road here.
1 John 4:2 (ESV)
2 By this you know the Spirit of God: every spirit that confesses that Jesus Christ has come in the flesh is from God,
The early church had trouble with this one right away. How were they to tell who was a follower of Christ? Among the very early church (the first handful of generations after the original disciples) articulating their faith in a way that distinguished them as uniquely Christian is what we know as a Rule of Faith. Wikipedia has the following to note on the subject. Understand that Irenaeus was writing about 120 A.D. and recording what was already a widely circulated Confession of the time.
In Christian theology, it is a principle which evaluates religious life and theological opinions by testing them for consistency against what has been firmly believed. The original rule of faith in the Early Christian Church as Irenaeus knew it, included the following:
… this faith: in one God, the Father Almighty, who made the heaven and the earth and the seas and all the things that are in them; and in one Christ Jesus, the Son of God, who was made flesh for our salvation; and in the Holy Spirit, who made known through the prophets the plan of salvation, and the coming, and the birth from a virgin, and the passion, and the resurrection from the dead, and the bodily ascension into heaven of the beloved Christ Jesus, our Lord, and his future appearing from heaven in the glory of the Father to sum up all things and to raise anew all flesh of the whole human race … :
Gradually, these Creeds grew in length. And for good reason. Beset with heresies as each new group and idea came along with some new and unique way to reinterpret Scripture differently than was understood by the early churches under the guidance of the Apostles, it was necessary to add statements on these issues clarifying what was orthodox belief and what was deviant. While sticking to the core details, these expressed in detail the specific things understood about the faith and the teachings of Scripture in a way that set Christians apart from people whose doctrinal systems had other allegiances.
For example the Apostles' Creed, known to have been coined by at least 215 A.D but has several variations, of which we typically use a little later more developed one, goes something like this:
I believe in God, the Father almighty, creator of heaven and earth.
I believe in Jesus Christ, God's only Son, our Lord, who was conceived by the Holy Spirit, born of the Virgin Mary, suffered under Pontius Pilate, was crucified, died, and was buried; he descended to the dead. On the third day he rose again; he ascended into heaven, he is seated at the right hand of the Father, and he will come again to judge the living and the dead.
I believe in the Holy Spirit, the holy catholic church, the communion of saints, the forgiveness of sins, the resurrection of the body, and the life everlasting. AMEN.
That covers a lot of little details about the person of Jesus, but groups still arose that claimed various teachings that subverted what Christians understood the teaching about Christ to be. The bit I bolded above from 215 A.D looks more like this in the Nicene Creed from 381 A.D.
[...] And in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all worlds; God of God, Light of Light, very God of very God; begotten, not made, being of one substance with the Father, by whom all things were made. Who, for us men and for our salvation, came down from heaven, and was incarnate by the Holy Spirit of the virgin Mary, and was made man; and was crucified also for us under Pontius Pilate; [...]
One could go on and on, tracing the development of longer or more specific Creeds and Confessions in response to heresies and misunderstandings. As the number of ways man invents to deviate from the teachings of the word grows so will the need to state clearly what the orthodox teaching on those issues is. However, I digress from your question.
Specific points of doctrine such as those you give as examples do not originate with the Confessions that most clearly express them, rather they originate with the Scriptures and are the articulations of orthodox interpretation which the church in various ages felt the need to articulate when faced with influxes of teachings that did not match its own yet claimed to be Christian.
One must be clear on this: Creeds and Confessions do not hold authority over Scripture nor is it the confessions themselves that we teach, but rather our interpretation of God's word using the wisdom of the church that has gone before us as a measure or guide to make sure we have correctly understood it. The WCF itself clearly states this in Article 31.42:
All synods or councils, since the apostles' times, whether general or particular, may err; and many have erred. Therefore they are not to be made the rule of faith, or practice; but to be used as a help in both.
Many other written confessions include a similar note about themselves.
I do mean all. Even sects such as the Bereans who insist on having no creed but the Bible, have as a commonly agreed upon statement that in order to make any doctrinal point, only the text of Scripture can be used. In effect this is their confession. Their collective interpretations also tend to have some unique flair to them do end up documented as statements of faith that set them apart from non-Bereans.
This is not one of the points to which I take exception.