The house is worth conservatively $400k (with no debt on it) - rent $2400/month.

With such a head wind coming from getting a commercial loan you may want to look at non-qualified mortgage products like debt-service-coverage-ratio loans or possibly a HELOC.

My FICO is around 740(2) Since my debt will increase significantly after the refi, I would assume that leveraging further using the $400K will be more challenging (and risky). 

Fanny/Freddy are out - already have 10 mortgages.  5 were just refinanced - not enough equity for cash out but lowered the monthly debt service by about $800/mo. 

Other options you may consider would be a 2nd loan or HELCO (they have higher rates in the 5s and 6s) or if you did the loan as an FHA you could refinance as a conventional loan and if you have over 20% equity you can eliminate your mortgage insurance and lower the monthly payment. 

I once thought that due to my situation I should not pursue anything but though mentorship I learned proper planning eliminates fear of what happens to everything once the time comes.

This was a primary residence for 8 years, a rental for only one or less.The good news is that it does qualify for the Section 121 Exemption, which is a better plan if you qualify. 1031 essentially passes the buck on the tax bill, Section 121 ELIMINATES the tax bill.It sounds like you qualify - you lived in it as your primary residence for 2 of the last 5 years (actually 4 of the last 5).

I’m looking to buy a 4-20 unit property, which the cash flow would definitely service the new debt. 

The rate becomes less important the shorter you keep the debt.

The refi amount will be limited by the relative contribution the refi would add to the debt side, and the NOI of the rent would add to the income side.