These are questions for your Title Company and your Lender:
We cannot possible answer them; but I'll try.
I doubt that any Bank will go 60 days on the first payment; I would bet that you would have to beg for 35-40 days, and base your request on when you get paid. Understand that no matter when it is due, you will be paying interest on it, increasing your Vig.
Your Lender should have given you your GFE, which should show everything. They must, BY LAW!
And your Title Company should have given you your HUD1, which shows everything else. You may ask your Escrow Officer to sit down and explain everything to you: They do that.
Note: If you CLOSE in January, you will be pre-paying 6 month's of Taxes, whereas, February drops to 5 month's worth.