They must be held in escrow by a third party called QI - Qualified Intermediary - and it must be engaged before you sell. 

Will the rental period be considered a non-qualifying disposition?

Once you have that you can qualify for a low money down loan and buy a double. 

This is a portfolio loan offered by local credit unions where you can buy up to 5 pure investment properties with as little as 5% down if you qualify, which you do have to qualify still with DTI, credit, reserve requirements, etc.There are different credit union options that would be best to connect with based on where you are investing and your situation, so that is why it's important to connect with our team to guide you on the right path on where to apply so you are not wasting your time and having multiple unnecessary credit pulls.

This is why they help you build your personal side first (if you need it) so you can better qualify to get business funding!

After a year living rent free I could have some money saved up again but still won’t be enough to buy a fixer upper outright and BRRRR (Without doing the math) and I wouldn’t be able to qualify for another FHA loan presumably.

You may have an issue with purchasing in Tampa if you do keep this one as you will be qualifying for a second home and your DTI may not handle that.

If I become a part-time real estate agent and have no other W-2 job, would I be able to qualify for real estate professional status?

Assuming the property is equally owned 4 ways (25% each) you could have the property appraised for confirmation of value: $1.8M/4 = $450k/share x 3 = $1.35M could be obtained as a cash out refinance, potentially utilizing a healthy portion of the STR or projected STR income to qualify. 

This will help you find out if you qualify way before submitting the loan.