Firstly, if a mech has two fists free, does that count as Two Weapons for the purpose of using the Mechidexterous Feat?
That never occured to me before. But yes, I think that would be OK. It fits within the description of the feat, and doesn't seem too unbalancing.
Secondly, if someone uses the Mechidexterous Feat, and attacks with two melee weapons, does he do only one half strength damage with the second weapon, like a person does when fighting with two weapons?
That's a good question. The d20 rule is based on the fact that you do 1.5x Str if you attack with a two-handed weapon, so attacking with two weapons in two hands should provide the same Str bonus. In the DragonMech rules, I don't think I mentioned anywhere that mechs get 1.5x Str bonus if they attack with a two-handed weapon (although you could make a case for it). I assumed that due to their mechanical construction, they get the same Str bonus no matter what limb they attack with. It's not like a mech can be left-handed. I'd be inclined to say that a mech gets its normal Str bonus no matter what melee weapon it attacks with, and no matter order in which it attacks with them. This makes them a little more powerful relative to monsters (which take the 1/2 Str bonus on subsequent attacks) but it's consistent among all mechs, so it doesn't unbalance them relative to each other.