What is the balanced chemical equation for carbon dioxide with hydrogen gas to produce methane and water?

You are referring to the Sabatier reaction.

CO2 + 4H2 ? CH4 + 2H2O

I will leave it to you to do the simple arithmetic for the yield.

Not sure if there are any side reactions with your first equation. If not its CO2+4H2 CH4+2H20

Combustion of methane has several steps.

"In the combustion of methane, several steps are involved:

Methane is thought to form a formaldehyde (HCHO or H2CO). The formaldehyde gives a formyl radical (HCO), which then forms carbon monoxide (CO). The process is called oxidative pyrolysis:

CH4 + O2 ? CO + H2 + H2O
Following oxidative pyrolysis, the H2 oxidizes, forming H2O, releasing heat. This occurs very quickly, usually in significantly less than a millisecond.

2 H2 + O2 ? 2 H2O
Finally, the CO oxidizes, forming CO2 and releasing more heat. This process is generally slower than the other chemical steps, and typically requires a few to several milliseconds to occur.

2 CO + O2 ? 2 CO2
The result of the above is the following total equation:

CH4(g) + 2 O2(g) ? CO2(g) + 2 H2O(l) + 891 kJ/mol (at standard conditions)
where bracketed "g" stands for gaseous form and bracketed "l" stands for liquid form."