wall mounted toilet leaking methane gas?

The first thing is to narrow down the possibilities. You said half bath, so I'm assuming there's a sink downstairs, is there also a laundry tub, in any case plug them off with a test plug. If the smell goes away, there's your culprit. Usually when you get sewer gas from flushing a large quantity of water, the trap water is being sucked out by the suction it produces. Have someone downstairs to watch while you run the water to see if you get motion in the trap of the toilet. If none of this is happening, you must search for a crack in your vent line. If anyone can go on the roof safely, put a hose in the vent and see if anything leaks. The fix for a sink is to put a flapper check valve in the drain leading to the sink, like the one's used on sump pumps. For the vent, if it's cast iron, a cheap repair is fiberglass repair kit. Just get everyone out of the house when you do it because of the epoxy smell. Good Luck!

It is very unlikely you would get gas out of your toilet anywhere but at the wax ring. Could the odor be coming from another source, like the sink, a floor drain, or the washing machine drain? It would obviously be better if you can nail down the source of the smell before you start trying to replace anything.

Its sounds like that wax o ring has disentegrated. This ring sits on top of the actual hole in the floor (opening of the sewer pipe) and the hole in the toilet.You are going to have to call a professional plumber. Start saving your money. Or get some plumbing students. Also be sure you dont have a sess pool that may need emptying.It really sounds like a blockage of some kind.Call your plumber.

Have you had your plumbing vents checked?

All sinks, toilets and tubs connect to a vent(s) that generally goes up through your roof.

These may be clogged with debris/snow.

If they are blocked, any air/sewer gas in pipes will be forced up the nearest drain as you drain/flush your water.

yep, snake your sanitrary vents, make sure the wax ring is good, and the problem will go away....