@Flavia Vangelotti With your expressed concern for cash flow, I suggest considering a Debt-Service Coverage Ratio (DSCR) loan.

Rentals loans/ DSCR loans are typically 20% down 

I have heard that paying off your rental is not necessarily the best choice because of the deductions it allows for, but given the type of loan this house is mortgaged with, is it even possible to buy another home together and start renting this one out when I have a FHA loan on this one?

I'd start by talking to a mortgage broker/loan officer to find out how much you're qualified for. 

They also receive information on operational expenses through one of their loan servicer partners but I don’t find that to be entirely accurate.

Though it is over 50% cheaperin my market, the price came in about the same although Reonomy isn't market-based but is limited with contact search, Reonomy has loan maturity data which co-star is lacking which is a big upside. costar has leasing data and is  also more accurate. 

Depends on your goals. refinancing is basically starting over with a new loan.

A bridge loan might be your best option here.

I have not secured financing at this point so this is all out of pocket and cost is a constraint. 

Which will be best for down payment and which loan to choose.