Go to re-light water heater after gas was turned back on and the plastic knob on top of the gas valve where pilot light is turned on is missing.  

Give them a heads up before doing treatment with what the pest guy says needs to be done,, trash, put food items in plastic containers,, ect.If they don't comply give them notice to move, non- compliance with terms of lease, cleanliness. this could infest your entire building, and I would have pest guy treat all neighboring apartments. 

Get a basic "handyman toolkit" in a formed plastic case.

Run out to Bed Bath & Beyond and get a bathroom trash can that a plastic liner fits in with a nice fitting lid.

Sometimes a simple change to the grade around the foundation can fix the issue or if there is a cement side walk/patio, make sure it’s sealed.

They don’t take up a lot of space, and people LOVE mixing up some cocktails & sitting around the fireplace at night (or patio) playing board games. 

Peanut butter on glue traps, check the traps daily or you will hear the little guy fussing and fighting to get free, use a plastic shopping bag or two as a glove to pick up and wrap in the glue trap with mouse still on it in the bag and tie that in a knot, then proceed to step on him or give him a fast swing and slam on the side of the trash can, then dispose of the dead vermin.The poison, is not always attractive bait and then they can die in your place and smell, or be eaten by other animals that we don't want to kill(owls, bobcats, pet cats and/or dogs)About a dozen kills to my credit.

Also, do you recommend increasing heated area to accommodate a third bedroom by converting areas such as patios? 

That doesn't mean it's always a train wreck; I've had tenants build nice brick patios and do other stuff like that, and hey great if it increases the property value or decreases your cost, but I would be real careful with that.

In pulling out the old one we discover that the tenant has stuffed plastic carrier bags all through the back of the stove and around the drawer at the bottom of the stove.