If I have 6 cards in my hand and use Mimikyu's "Impersonation" attack to discard Erika's Hospitality from my hand, can I use the effect of Erika's Hospitality?
Since you have 5 cards in your hand after discarding the Erika's Hospitality you can use its effect.
Source: TPCi Rules Team (2023-04-06)
If I use Mimikyu's "Impersonation" attack to discard an opponent's Whitney, does that Whitney count for the "draw 2 cards for each other Whitney in your Discard Pile" because you're not actually playing that Whitney, just using its effect?
No, you still cannot count that Whitney towards the number of cards to be drawn because its text only includes any OTHER Whitney in your Discard Pile.
Source: TPCi Rules Team (2019-12-05)
When using Mimikyu's "Impersonation" attack, can I discard a Supporter like Green's Exploration or Morgan and bypass the requirement that says you can only play the Supporter if you do such-and-such?
You can discard those types of Supporter cards with Impersonation, but if you don't meet the Supporter card's requirements you cannot do what is says.
Source: TPCi Rules Team (2019-12-05)
If I use Mimikyu's "Impersonation" attack, do I get to use the "when you play this card" part of Tag Team Supporters or not?
Impersonation gets to use all effects of the card; you still do the entire part of the card even if you're not "playing" it (but you still have to pay any required costs).
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
If I'm not allowed to play a Supporter card for some reason, can I still use Mimikyu's "Impersonation" attack to use the effect of a Supporter?
Sure. You are not playing the Supporter, just using its effect.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)