Go to city building & assessor offices to see what you can dig up about the property, including past permits and inspections, tickets, etc.Make sure you get all tenant contact info and application info at closing.

Did he handle getting all permits, plans drawn, etc.?

When all factors  are considered in totality, the applicant's 2.7 income factor may be a "safe" risk, but if the other factors indicate a problem with this applicant's qualifications, he may prove to be riskier than you should permit.

Quote from @Robert Ellis: new builds set the price premium in every zip code I've ever studied. in addition new builds don't take 9 months. we just built a house in 60 days. we wait till permits are approved and shovel ready, investors buy land cash, entitle, contracts, negotiate with financing, place financing when we are ready and break ground. but we build the same over and over and don't go outside our Floorplans 60 days from Dig date maybe.

@Louisa DavisGenerally, I find meetups are powerful for relative new comers to real estate.  

As @Bill Brandt mentions it's probably a bad idea to permit Great Danes in a 500 foot walkup apartment, so adjust as appropriate.

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.

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.