Hard to get your exact scenario right without the extra info.

E.g. let's say some of the properties have an extra bedroom/bathroom, finished basement, newer/older homes, etc.Thanks!

That gives you a couple extra days to recover and not pay late fees.

If I did this cash out refi, I would replenish/add to the emergency fund just for a little extra comfort but would like to use the rest to buy more real estate. $40k wouldn't buy much around me so likely I would have to look out of state which I have never done before.  

That's only an extra $50-100 per month?

However, they've earned themselves a bad reputation (and a lot of it is deserved).

The extra 10+% is well worth the bit of effort needed to self-manage. 

Without that extra protection, Landlord risks getting sued personally.

The reality is the opposite - if it's not in writing then the PMC doesn't have to provide the service or can charge extra for it.We have a 12 page management contract that we've added our real experiences to over the years, with the intent of protecting both us AND the landlord.

I don't recommend it, but if you have extra cash on-hand, you can waive appraisal if you are confident in the value of the property long-term because cash has no appraisal contingency.