If I copy Muldrotha, the Gravetide with Protean Thaumaturge and there’s a Mirror Gallery out, then I cast an enchantment (no other types) from my graveyard using the Protean-Muldrotha, 1) can I choose to re-copy Muldrotha with the Protean, and 2) if I can and do, can I use the Protean-Muldrotha to cast another enchantment (with no additional types) from my graveyard?
When it copies Muldrotha it gains a completely new although identical identity. This new identity has an ability that grants permission to do something. It doesn't care that it previously had a similar ability in which is used up its permission.
Note that if two or more effects allow you to play a land or cast a spell in a manner not otherwise allowed — e.g., if you control two permanents that have Muldrotha's ability — you should announce which such effect to make use of as you play that land or cast that spell, in case of ambiguity. This can be relevant if, for example, Mirror Gallery leaves the battlefield and the legend rule kicks in again (review C.R. 704.5j, 101.1). See also this thread.
EDIT (May 5, 2020): Edited in view of update with Ikoria: Lair of Behemoths.
EDIT (Jan. 6, 2020): Edited.
Is this due to rule 706.2, perhaps? Could you give me the rundown of exactly why it works? I’m simply asking because one of my friends who is also a judge is unsure of the interaction and it’s a tricky one to get, so hence the initial question anyway, haha
It is simply due to how continuous effect are applied (the layer system). Within layer 1, copy effects are applied in time stamp order. Each time the Thaumaturge's trigger resolves, a new copy effect is created, which overwrites all the previous ones because it is the one with the newest time stamp (the others are still there, they just have no visible effect on how the permanent looks in the end anymore). It doesn't matter, that the copiable values become the same as before each time, it is still a new effect, and thus gives the Thaumaturge a new ability, one that has not yet been used to cast an enchantment from the graveyard.