Why do people at the EF say the St Michael prayer after Mass? Is it a prayer of thanksgiving?
Because the St Michael prayer was imposed by Pope Leo XIII among others, such as the Salve Regina, for the whole Catholic world to pray for a settlement between the Papacy and the kingdom of Italy. Leo XIII's predecessor, Pius IX, lost the Papal States during Italian reunification in 1870. Leo XIII spent his entire pontificate as a "prisoner of the Vatican" (i.e. he never left the Vatican for the rest of his life after being elected pope as a form of political protest). After the Vatican made peace with Italy through the Lateran Treaty of 1929, establishing the present-day Vatican City State, the intention of the Leonine Prayers was changed to praying for the freedom of the Catholic faith in Russia, then the Soviet Union.
The Second Vatican Council suppressed the Leonine Prayers, so they are not strictly of obligation for TLM's of any kind today. But TLM communities will continue to recite them after low Mass because this was the common practice before Vatican II, and they are well-liked. Trads who are aware of the Leonine Prayers' original purpose will broaden the intention to be for the conversion of Russia, or for defense against the "errors of Russia" (such as international communism).
It is my understanding that at a Missa Cantata (correct term?) it is not said. Why?
Pope Leo XIII's directive for the Leonine Prayers only applied to "private Masses". At least in the US, this was interpreted to apply to the literal sense of
missa privata, the technical name for a recited Mass (low Mass). In the US, low Masses were overwhelmingly the norm even for parochial Sunday worship, so this meant almost all Masses used the Leonine Prayers. I believe Leo XIII's intention was more limited: that the Leonine Prayers would be said at the end of Mass of a private character, such as weekday liturgies offered for private/personal intentions. Not Sunday Masses, weddings, funerals, etc. These occasions would be when a Mass would be most likely to be sung.
At a Missa Privata (low mass) people only recite the Salve Regina? What is the purpose of not singing it?
If the whole Mass is recited, it makes sense to recite the Salve Regina too, instead of sing it. Singing the Salve but not the Mass would seem to suggest that the post-Mass devotions are more important than the liturgy itself.
Also, why do Salve Regina have melodies when St Michael prayer and Ave Maria prayer do not have melodies?
The Salve Regina is an antiphon. It was written explicitly for the purpose of being sung.
The Ave Maria does have chant melodies, as does the Pater Noster. However, these prayers were also memorized and recited, even in Latin, by the lay faithful for most of Church history to a degree beyond that of the Salve Regina.
There is actually also a chant for the St Michael prayer, but it's difficult, obscure, and certainly wasn't originally written by Pope Leo XIII with music in mind. Nevertheless, I have sung it a few times.
