I ended up interviewing a brokerage prior to starting my online courses.

Secondly, I have documented violations of the lease prior to them moving out.

THe prior owner used a property manager. 

Look at what your Interest rate is currently compared to what you can get now. and then compare that to what you are saving by removing the PMI.

We will not be closing on our new home until August 15th and have some items we plan to do prior to moving in. 

I have NO prior RE or investing experience but both seem like something I could see myself absolutely enjoying.

The effort I'd have to put in the 2 properties just wasn't anything worth while compared to the 240k property. 

If the prior owner just sealed up a doorway and called it a 3-unit, you’d need to get zoning approval, probably some inspections, and it’ll take time and money.

Find out how much the city utilities and taxes are compared to property prices and rental prices (craigslist is useful for checking rental prices), and then see where you want to start investing with the best return on investment. 

She is still under the prior lease, which you inherited.If she still doesn't leave or pay within that timeframe, then determine if you want to try evicting her yourself or hire a company/attorney to do it.