Almost all of this information has been discussed previously many times.

Mutianyu is much closer to the airport, and less than an hour away (by car) if no traffic. I've never made it to the GW by public transport, but making your way to the train station (BJ West if I remember) for Badaling is rather a trek. The bus to Mutianyu is closer, although the total journey time may be similar.

You need to plan on arriving at the airport about 90 minutes before your flight. I wouldn't cut it much finer, although I've done an hour comfortably enough if the security queues are not too bad. Being in C doesn't get you fast-track at security.

I'm not sure what you mean by 'tourist trap' -- all the touristy (i.e. non wild)sections of the GW will be filled with err..tourists. Much like any other major world attraction.

You appear to be cost-averse, in which case, come late morning and leave evening, but get a car to Mutianyu. That will cost less the same as an overnight (or less), which would be the only way I would recommend getting a bus or train to the GW.

tb