You do need the plumbing tested, and obviously you need to know where the water is coming from. 

After the house flooded a second time with raw sewage the second plumbing company found mud in the pipe the first plumber did not clean out.

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.

How's the plumbing system?

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!

Cash or finance, what of the numbers you ran on the property (taxes, insurance, albeit outdated what of the major systems you might have to fix or replace like roof, hvac, plumbing, electrical). 

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 am very handy but have never tackled flooring, cabinetry, countertops, minor plumbing etc.  

I have been wanting to do this since the early 2000's, but life kinda got in the way, plus the late 00's were not too great for new construction (my main plumbing gig at the time) Which is when I lost my first nest egg...