The property manager gave the tenant a 60 notice and now the 60 days are up. 

Or, if you need a few weeks to find a less expensive place, we will let you out of the lease agreement with 30 days notice. 

Also, finding a bank to do 20% down on a house that cheap will be tough. 25-30% down will be more common.

Useful articles:The Inside Track to Vetting A Multifamily Deal SponsorHow to Avoid the Most Common Commercial RE Due Diligence Mistakes

The person in the basement almost started a fire which caused the fire department to come out, and when they noticed the third unit they threatened the owner with $10k plus in fines.

But, of course common sense says to aim to buy below retail if possible.

If you've owned the house long enough that rent was due 2 1/2 weeks ago (which was also after your closing), then you should have given her a 3-day or 5-day (whatever it is in your jurisdiction) Pay or Quit Notice. 

I also notice that he’s managed to get everyone (me included) to say yes about 30 times and it’s not 10am yet.

I'm not sure I understand the question, but as long as you give the current tenant the legally required length of notice for the non-renewal of the lease, after they move out, someone else can move in immediately.Edit: if it's clean and move-in ready

Yet I have seen it time and time again.Even if they're supposed to leave at the end of the month, I'd still file a 3 or 5-Day Pay or Quit notice (whatever it is in your jurisdiction).