Should I, and do I need to, write a letter to the bank to request permission to live elsewhere and rent out all the units in the building I just purchased?

Go to many different banks and talk to the loan officers about what they can do for you.

I needed his Bid ASAP since I am using a 203k loan for the rehab and the bank said it needed his bid before they could proceed with the appraisal.

Try shopping around a bit more, I tend to prefer regional banks vs. local banks for this type of property.

“Yes, I track my money by reading my bank statements daily.”4.Do you feel like you have a good idea of exactly where your money goes?

There are some portfolio lenders that will do a 24 month bank statement for self employed or rental cash flow.

To answer your question,1) let me know if this is what you were trying to achieve.The easiest and simplest way to do this is:  - have each property(or few more depending on your risk threshold and situation such as different LLC for different State) under one LLC and have a rental contract for houses under the same LLC, keep a separate bank account and book for this LLC. and repeat the same with other LLCs. 

Banks have their place , but I try to avoid them whenever possible.

Talk to your local bank/credit union to see what type of financing you qualify for.

I specify in my ad that an excellent credit score and income is required as well as documentation (paystubs, bank statements, references, etc) and background check (costs about $25-30)I rent in a B neightborhood, I charge above market rent for the area but keep the place in top-notch shape.