So, if a tenant pushes back on the MTM, to me that's a red flag, because there really is no down side for them unless they are up to no good.

Depending on which region, you're probably going to have appraisers that look at dollars per square feet averages and then adjusting numbers based on red flags that are far outside the norm for the area you're coming in.

Paying with a money order is a red flag that they don't have money in the bank.

It appears to be a major red flag to me and I'd like to negotiate to have them remove it or find another company.

Red up on wholesaling. 

If it happens again, red flag, and they will have 30 days to leave and/or include that you have the right to  do progress checks with their Program Advisor for the first 90 days once a month.

Not trusting your agent 100% is a bit of a red flag, if you don't feel like they have your back at all times, you should find a new one. 

If the inspector does not allow you to join, that is a red flag to me. 

sound like a form of reverse red lining..

So it's really two sides, look for those red flags but ALSO look for those Green-flags also, look to see who they are as a person, why are they doing it, are they passionate about what they are doing and guess what there is a passionate person about everything out there, my insurance guy LOVES insurance, lol, seems nutty to me but he absolutely lives, eats, breathes insurance.