I'm certainly in favor of Miguel and Greer having some getting-to-know-you time. :)
I figure Miguel wouldn't be able to refuse Tigra's request, and he'd spend the night with her at the Throwback's safehouse. But after that he'd head back uptown and make some arrangements to sneak her up to his apartment, where he can set her up with an image inducer to hide her appearance. This being the same kind of image inducer that Nightcrawler and Beast have used when they want to be incognito in public; I figure by 2099 it would be further miniaturized and fairly commonplace. And then when she and Spider-Man want to go out as a vigilante duo (someone's going to mistakenly call her Black Cat, lol), she can just drop the disguise. Maybe he gets her a job as his lab assistant or something?
And yes, Greer and Lyla would make a great comedy duo. Greer's sass and Lyla's innocent wit. And it'll be fun to play Greer off of Miguel's exes Dana D'Angelo and Xina Kwan, his brother Gabriel, his boss/father Tyler Stone, and other members of his supporting cast (like Kasey Nash, pictured in the avatar).
As you might have guessed, I had Hobgoblin 2211 escape into 2099 as a possible recurring villain. This gives Miguel his first "Goblin" adversary, because Goblin 2099 wouldn't happen until a later point in Miguel's history. I could see Hobby investigating some temporal means of trying to get back home, and giving Miguel and Greer more villains to fight as a distraction. Sinister Six 2099?
___ I'm guessing Kitty's move at this point is to want to see if she can get together with Jubilee on the sly? And then it's just a matter of actually being able to spit the words out without being interrupted. By EVERYONE.
Seriously, the most fun part of the mansion party is that there are so many character combinations we could experiment with. Like all the fun trickster characters from the various teams combining forces. Or the big tough guys having an impromptu drinking contest. Or somebody daring Guido to cannonball into the mansion's pool and splash the mansion.
Speaking of which: opportunity for Kitty and Jubes to check each other out poolside! Or during their attempt at a conversation, Jubes sees some shenanigans outside at the pool, and she wants to be a part of it. She invites Kitty to join her, they go up to Jubes' room to get changed into swimsuits, and then they can get get sidetracked by actually having the conversation. How's that sound?