The precise remedies available will depend on your lease and the contract law governing it.Since you are remote, you will need to find a property manager or someone local to be your boots on the ground through the drama which may ensue. 

You do not need more leverage, you can reject any applicant for any reason as long as it does not violate the protected species regulations.

If im doing private showings of my home, is there any discriminatory laws or regulations etc for requiring buyers to have a pre approval letter prior to showing?

There are often regulations on where and when you can place signs, if at all!

Make sure that your property manager is a licensed Realtor or Broker, look them up on DBPR, the department of business & professional regulation.2.

I moved into a neighborhood governed by an HOA in May.

Typically it is achieved via zoning regulations, like R1 (single family homes) vs R2 ( MFUs). 

2 - Are there any rent control/ stabilization regulations making the eviction impossible, like in NYC?

They are government owned foreclosures that are open to owner occupants before they are available to investors.

Remember, whenever you hire someone on a regular basis (W2), you face State Employment Regulations: { ssi, workmans comp, tax withholding }.