What is a good squad composition?

I always go with 2 shieldsmen, one or two snipers, a shotgunner, a machine gunner, and the rest riflemen, and I pick them with the following process:

• Anyone with 70+ strength can be a machine gunner. (This is uncommon at low levels, but you should have at least one in your initial group.) Pick your most accurate strong xenonaut, give him a machine gun, and set him aside. More machine gunners are just not that useful; the machine gun costs 80% TU to fire, so even a very fast xenonaut can fire, stand, and only move about 3 squares before kneeling again.
• Anyone with 65+ accuracy can be a sniper. I've found accuracy lower than this makes sniping kind of pointless. Pick your fastest accurate xenonaut (highest TUs) and make her a sniper. Take another sniper if you have another extremely accurate but low HP soldier, but I've found that Xenonauts maps are generally not open enough to really make snipers as overpowered as in new XCom. Unless of course you decide to commit a completely unconscionable amount of collateral damage. In which case, more power to you! Like machine gunners, snipers are largely unable to fire and move on the same turn. Sniper rifles (and machine guns) lose accuracy if the soldier moves before firing on the same turn.
• Scouts need to be fast (high TUs). Find your 3 fastest remaining xenonauts. The highest HP among them gets a shotgun and an assortment of grenades; the other two, as long as they have at least 55 strength, get shields, stun batons, and pistols.
• Everyone else carries a rifle. Rifles are light, so these soldiers should be able to carry medkits too.

Make sure to have every single soldier at his maximum carry weight without slowing down, so that they train strength as they run around. Give them more grenades until the encumbrance bar has a red bit at the end, then take the last one off. The purple gas grenades are absurdly good.

As to your last question, you're not being graded on how fast you complete the mission, except on terror missions. That should answer that question. Don't split the party. Don't Split The Party!! On terror missions, consider bringing an extra shotgunner; they're really unmatched in tight spaces, and shieldsmen are really worthless against reapers anyway.