Have the tenant use ACH or Zelle for deposits. 

You picked the wrong kind of property to start in as a 3/3.1 townhouse is not ideal for room hacking and is more residential in nature which is why you should sell it and take the proceeds and deposit money and look for a two-family where you and your brother can live in one side and you can full rent the other side - maybe even as a mid-term rental to 2x the rent.

So is forming an LLC and getting an EIN to set up a business checking account to receive the security deposit and collect rent payments a good idea?

My PM charges a pet fee that they keep, but they also guarantee if the pet causes damage (and the deposit doesn't cover it), that they will fix it. 

Flooring can be a basic vinyl or even a DIY job using peel & stick flooring.Expensive materials is more risk for the landlord because when they get damaged there is more likelihood you will lose money on the deal because collecting damages exceeding deposits if difficult.

Does the security deposit go to the new owners at closing, and does the tenant need to sign a new lease agreement?

There are plethora of landlord platforms that force you to use their own 3rd party banks to deposit rent and charge $2.5-$5 per transaction to directly ACH into bank of your choice (ex: I love Chase Bank).

This protects you if the tenant tries to tell you that he prepaid rent for the next 6 months or he had a huge deposit on file.

You also have them pay a deposit just like long term rentals.  

This will help you to communicate with the tenant regarding maintenance requests, demands for service, handling maintenance and repairs during the tenancy, what the tenant may owe during the tenancy, handling security deposits and claims, etc. 4.