Would waiting some period of time make that safer?

This tenant takes care of the unit she's in now, but she always pays late (sometimes incurring a late fee, sometimes within the 5 day grace period).

They informed us that we were past the inspection period becuase fannie mae starts the countdown when the "verbal" agreement is made, not when the final contract signatures are sent. 

Your option over a period of time can be cause for a lender to exercise the due on sale, as to the end byer paying it off at the end, really no issue, the loan is to be paid off at settlement, the lender won't be involved except in giving a payoff and settlement instructions to accomplish the payoff.

Small local banks in my area do not require a seasoning period.   3) Proof-of-funds: It's best for all involved to a have a pre-approval letter.  

However if there aren't enough comps that fit those constraints (usually we want to find at least 5 or 6 similar comps) then we have to either expand the search area or the period of times (but no more than 1 year) when Im performing an unofficial BPO (aka ARV) for myself or an investor friend/client I will include pending properties.

Amortization Period?

The latter option would save you the out of pocket expense while effectively financing the cost over the period of the loan.  

Contingency part : Is it better to put cash and cross out all mortgage contingencies or put mortgage 95% LTV etc and give myself 30 days “to obtain a mortgage aka finding a cash buyer” If I put cash they might ask about proof of funds ..and I also limit the time to find a buyer ..as attorney review period is only 5 days in IL.

@Matt Laird REOs generally have a OO period.