We increased the rent slightly after the first 24 month lease and were able to recently increase the rent when the first tenant moved out.Lessons learned?

It's a short term lease.

If this is simply you not wanting to accept partial payments and the rent is current, stop accepting it - and send a Lease Violation/Notice of Quit for Non-Payment of Rent - and force the full rent to be paid per the lease or start eviction.  

Maintenance at at PM company or one of the big hedge owners like AH4R, leasing at apartment complex will teach you a lot about applications, fraud, leases, collections, evictions. 

Is this a 2 bedroom apartment that is currently being leased (in the eyes of the CHA) as a 1 bedroom apartment?

What are some things I should include in the lease?"

I also consider this entire Arbitrage / master lease as passe'.

If you do, I recommend you hire a real estate agent to help you lease up the units.

Two tenants leased the 2BD/1BA, and the studio ran on short-term rentals.

Your lease probably specifies the residents must comply with "quiet enjoyment" of the property.