Water – What type/thickness of drywall should be used in water heater closet

The amount of oxygen consumed by a hot water heater is negligible. The risk of the gas heater sucking all the oxygen out of the room is zero. Now that does not mean that it's code to have it in a bedroom, but there is no safety issue from consumption of oxygen.

The safety issue is on venting of the carbon monoxide (CO) from the hot water heater. In order to make the install safe, ensure that the new gas water heater is properly vented through the ceiling. There are lots of resources online about properly venting gas appliances. Here's a reasonable one to start with.

Obviously since a gas hot water heater requires adding a vent, it's more expensive to install. You may find that an electric water heater still makes financial sense once you consider installation costs.

As noted in the comments, be sure and get a CO detector for the room. These are relatively inexpensive and you can get them at your local hardware store or order through Amazon.

Personally, I would just build a rectangle out of 2x4s slightly smaller than the hole, lower it into the hole flush with the existing framing, then screw it to the existing framing. That will give you a nice surface to attach your new drywall to. Depending on the size of the hole, you might want a cross member in the box as well so your drywall isn't spanning too big of a hole unsupported.

As for cutting back the existing drywall - I wouldn't bother with that for a couple reasons:

First, you're likely to run into issues with the nails or screws that were used to originally attach the drywall to that framing. If you cut around those, then the old drywall isn't going to have a very secure connection to the framing unless it was glued. And if it was glued, then you have the problem of the glue residue getting in the way of the new piece you want to put on.

Second, the damaged areas of the drywall are well within the areas that you'll need to mud over anyways - figure on mudding out at least 6-12 inches on either side of the joint. So you're really not saving yourself any work by cleaning that up.