What is the practical process for getting permits for rehab properties?

These are usually legal for an owner to live in (as long as the bathroom is permitted).

You can increase the rent by whatever amount is otherwise permitted by law.

She pulled permits and lives there full time now!

If you are going to engage an architect and do full plans and permits, then you will likely need to do some pretty serious analysis.

Many lenders will require seasoning (ownership interest) for a minimum of 6-12 months to utilize a new appraised value, but many will permit a cash out refinance using the purchase price anytime after closing.

If you take over the company that has good cashflow that's a good indicator but it would be good to vet those properties out by making sure they have the appropriate permits to operate legally. 

I have some work needing done that requires permits and the contractors aren't licensed in the city where the permit is needed.

What can we do, I am currently in a contract with this property, it is originally a single-family home and the owner added (3) 1 bed 1bath units at the back (without securing permits).

Assuming I had the experience, I would only buy a place with structural issues if I had 1) an experienced and trusted structural engineer telling me a clear, and verifiable explanation of the cause(s) of the problem and a viable and affordable solution, 2) trusted and experienced tradespeople who would be implementing the solution (carpenter, mason, etc.) confirming with me that they could fix the issue (with 90% or higher confidence that the issue could be resolved within budget), and 3) confirmation from city inspectors that the issue (and an inability to fix the issue) would not result in condemnation/revocation of occupancy permits. ...and even then, I'd only proceed if I had the cash reserves to ride out a major series of unexpected problems (e.g.; time overruns, budget overruns, unexpected revocation of occupancy permits from the city, etc.)