What gage wire do you recommend for each run?

 

If you already have the wire, go with the 10 gauge, it won't hurt anything. If you're going to buy some wire, I'd go with 10 or 12 gauge for the brake and ground wires and 14 gauge for the lights. If the wire is too small for the brake actuation, you will get heat from resistance but not enough current to operate the brakes properly. 16 gauge might work, but I always like to over build just a scooch (<------fancy technical term) for a safety margin.

A heavier gauge won't add enough weight to reduce capacity. I'm not big on spending more money than I need to either. My Scottish ancestry makes me rather chea...er...thrifty.

 

If you plan on making home runs to everything to avoid all splicing, then you can use a much smaller gauge of wire since they'll carry less than half of normal current. I wouldn't run any home run grounds, I'd just use the trailer for ground.