There was Eru, the One, who in Arda is called Iluvatar; and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made. (Ainulindale)
There's absolutely nothing in Tolkien's writings to suggest that any Ainur were made at any other time - in fact there is a further reference to "all the Ainur" later in the Ainulindale. Since the Ainulindale is feigned to be a later work by Elvish sages, it must be assumed that "all the Ainur" still holds for that later time too.
Here I give the title page of Ainulindale D (published in Morgoth's Ring) to establish it's status as a later work:
The Music of the Ainur
This was made by Rumil of Tuna in the Elder Days. It is here written as it was spoken in Eressea to AElfwine by Pengolod the Sage. To it are added the further words that Pengolod
spoke at that time concerning the Valar, the Eldar and the Atani; of which more is said
Regarding the offspring of an Ainu and an Elf or Man, at the end of the 1937 Silmarillion we have Manwe saying the following:
Now all those who have the blood of mortal Men, in whatever part, great or small, are mortal, unless other doom be granted to them
This can conjecturally be generalized to a rule that all offspring of a union between a greater being and a lesser being are held to belong to the species of the lesser being; but like I said - conjecture. However, the evidence (of Luthien being counted among the Eldar) fits the conjecture.
Update: 1st March 2015
Footnote 53 to the Shibboleth of Feanor (History of Middle-earth 12) has this to say about Melian:
Melian alone of all those spirits assumed a bodily form, not only as a raiment but as a permanent habitation in form and powers like to the bodies of the Elves.
This is of course very late writing and other parts of it are in disharmony with more developed concepts elsewhere, but if we accept it then it's definitive: when Melian took her "worldly body" she became, to all intents and purposes, an Elf.
There is, however, one interesting case of one of the people of the Valar who was not created at the beginning, and that's Turin Turambar. The Second Prophecy of Mandos (not in the published Silmarillion, but given most fully in HoME 5) ends with:
But of Men in that day the prophecy of Mandos doth not speak, and no Man it names, save Turin only, and to him a place is given among the sons of the Valar.
This idea was never actually abandoned by Tolkien himself, but was editorially removed from the published Silmarillion based on conflicting text elsewhere.
The use of the phrase "sons of the Valar" is significant here, and it relates to an older conception, that of the "children of the Valar"; here I'll quote from Annals of Valinor in HoME 5 (note that it's important to distinguish between Valar and Ainur: the Valar are a subset of the Ainur):
And with them also were later numbered their children, begotten in the world, but of divine race, who were many and fair; these are the Valarindi.
This conception was abandoned during revisions to the Silmarillion after LotR was completed, but it certainly did exist in the earlier versions of the mythology.