Have Gas Water Heater...Wondering If It Needs to be in a Closet
The closet is protecting the water heater from damage. I don't think you should demolish it at the present time. I doubt leakage of unconditioned air into this heater closet is having any significant effect on your electricity bills.

It is my understanding that there is a code requirement in Dallas that tank water heaters have a pan under them, and there must be a drain from the pan to the crawl space, to the outside, or to a garage where it would drain out. But if you don't have a pan, I don't think you are required to remove the tank and install one. 

For the life of the current tank water heater, forget about about a tankless water heater for this house. The cost would not make sense for this dwelling. 

The first thing to do would be to get carbon monoxide (CO) detectors installed in the living space of the house (bedrooms and hall or living room). Next insure that the combustion gas flue meets code.

Then attack the rodent problem with snap traps (no poison) in the heater closet and in the attic if it is accessible. If you have children or pets, you must put the traps where they cannot get caught. If rats are in the attic or garage, they will find a way into the living space of the house. If you want to put traps in the garage, put them at a height where pets cannot get to them or put the traps in a box with an opening that denies entry to dogs, cats and children.