What is the oldest hymn (with music) that is still sung in churches today?
The simple answer to the first part is the Sub tuum praesidium
1784 processional banner of the Lisbon Holy House of Mercy depicting the Virgin of Mercy protecting all social classes; the first verse of the hymn is quoted underneath.
"Beneath Thy Protection" (Greek: Ὑπὸ τὴν σὴν εὐσπλαγχνίαν; Latin: Sub tuum praesidium) is a Christian hymn. It is the oldest preserved extant hymn to the Blessed Virgin Mary as Theotokos. The hymn is well known in many Roman Catholic, Eastern Orthodox and Oriental Orthodox countries, and is often a favourite song used along with Salve Regina.
The earliest text of this hymn was found in a Coptic Orthodox Christmas liturgy. The papyrus records the hymn in Greek, dated to the 3rd century by papyrologist E. Lobel and by scholar C.H. Roberts to the 4th century. According to scholar Serafim Seppälä "there are no determinate theological or philological reasons to reject the 3rd century dating."
The hymn is used in the Coptic liturgy to this day, as well as in the Armenian, Byzantine, Ambrosian, and Roman Rite liturgies. It was part of Sulpician custom that all classes ended with a recitation of this prayer. Besides the Greek text, ancient versions can be found in Coptic, Syriac, Armenian and Latin.
Henri de Villiers finds in the term "blessed" a reference to the salutation by Elizabeth in Luke 1:42. "Praesidium" is translated as "an assistance given in time of war by fresh troops in a strong manner."
The former medieval and post-medieval practice in several dioceses, especially in France, was to use the Sub tuum as the final antiphon at Compline instead of the Salve Regina and in the Rite of Braga where it is sung at the end of Mass.
An Egyptian Papyrus of the Third Century
The Sub tuum praesidium is probably the oldest Christian prayer dedicated to the Blessed Virgin Mary. This prayer was long used in both Eastern and Western rites, even if numerous variants existed at the time. In 1917, the John Rylands Library in Manchester managed to acquire a large panel of Egyptian papyrus -- the exact area where they were discovered is unknown -- including an 18 cm by 9.4 cm fragment containing the text of this prayer in Greek.
C.H. Roberts published this document in 1938 (cf. Catalogue of the Greek and Latin Papyri in the John Rylands Library, III, Theological and literacy Texts, Manchester 1938, pp. 46-47). Roberts then dated this piece of papyrus back to the fourth century, thinking it was impossible to find an invocation to the Theotokos before this century (we will however see below, that the expression Theotokos was already in use in Alexandria before 250).
However his colleague E. Lobel, with whom he collaborated in editing the Oxyrhynchus papyri, basing his arguments on pure paleographic analysis, argued that the text could not possibly be older than the third century, and most probably was written between 250 and 280. A contributor to Roberts, H.J. Bell, even said that this document might be a "model for an engraver" considering the beauty of the uncials. The Sub tuum praesidium thus precedes by several centuries the Ave Maria in Christian prayer.
In the Byzantine rite:
The Sub tuum is sung during Vespers in Lent in the middle of the final prayers after 3 troparia: the Ave Maria, a Troparion to St. John the Baptist, and a Troparion to the Holy Apostles. This place assimilates the text to the role of an apolytikion troparion which changes each day during the rest of the year. The apolytikia troparia are related to the singing of the Canticle of Simeon, which begins with the words in Greek Νῦν ἀπολύεις (Nunc dimittis). It is very likely that this series of fixed troparia at the end of Vespers during Lent represents an old state of the rite. Variable troparia were probably substituted for them for other days of the year. Moreover, the Horologion Grottaferrata seems to assign them at the end of ferial Vespers also during the year (Horologion, Rome 1876, p. 104).
In the Russian tradition, the Sub tuum praesidium is often sung for devotion, even outside of Lent, with the addition of the invocation "Пресвѧтаѧ Богородице спаси насъ" ("Most Holy Mother of God, save us") added to the end. Russian believers are very attached to this troparion. Parishes still widely use the text that predates the liturgical reforms of Patriarch Nikon in 1586; this fact is a clear sign of the strength of this attachment (such an attachment to the pre-Nikonian version is not observed for any other famous pieces of the repertoire - for example the Easter Troparion or "More honorable than the Cherubim".
In the Ambrosian rite:
In the Ambrosian rite this piece is sung as the 19th antiphon of the procession of the feast of the Purification of the Blessed Virgin on Feb. 2, a procession of 21 antiphons, many of which are originally Greek. It's music is similar to that of a Roman second tone. The 20th antiphon of the procession, that follows, presents a text quite similar to the previous one and is built on the same melody:
Still in France from the nineteenth century onwards, the Sub tuum is frequently used for benediction of the Blessed Sacrament. The Sub tuum is often associated with the Litany of the Blessed Virgin, as, for example, in the ordo of the procession made for the vow of Louis XIII in the proper of the diocese of Paris. Many old French liturgical books present the Sub tuum in a beautiful plainsong melody of the tone II. Here it is, taken from an edition of Digne of 1858:
The Sub Tuum Praesidium
Listen to the Sub Tuum Presidium on this YouTube video: Sub Tuum Praesidium (Feasts of BVM, Antiphon)