VP Offered Coffee or Lunch; What should I take
I personally think I'd do better in a coffee setting, but I think lunch may offer more face time. I also don't want to offend her by taking coffee when she offered lunch as if I'm too busy.
You're overthinking the hell out of this. Either is fine.
Order coffee with your lunch. That's the true meaning of Covfefe
Time to put on your big boy pants and counter with dinner at a fine Italian restaurant. Check beforehand to make sure someone playing a violin can approach your table during the "meeting" to serenade for an awkwardly-long period of time. If you see her frantically trying to downplay the situation -- that's good! Smile confidently and playfully. She's becoming intimidated and slightly turned on by your extreme alpha approach.
That's when you ask her when you can start. Before she says anything, tell her you'll think about it, place down your napkin and make an exit, but not before throwing down too much cash on the table for both meals, tip, and her cab ride back home.
Congrats ahead of time on your new job.