What is a good squad composition?

Classes

Shield (Shock Baton + Shield + 10KG Armor (Jackal or Buzzard): My shield users act as both scouts and general utility. I don't bother giving these soldiers pistols since if I'm close enough to an alien for a pistol to be effective I'd rather just whack them over the head with a Shock Baton (a lot of enemies can be taken down with 1-3 hits) or toss a grenade. You'll want these soldiers to have a high TU and high Strength.

Assault (Shotgun + Wolf Armor): Shotgun wielders, provided they can get close to an enemy, can take out most enemies in 1-2 hits. the key to safely approaching with a shotgun wielder is to suppress enemies first, and then have your shotgunners go in for the kill (Flash Bangs and Rapid Fire is good for this). If your Assaults have high enough reflex you might not even need to worry about reaction fire, since aliens will often fail their reflex test. Sneaking up on an alien from behind is another effective way to approach them, since they can't take reaction fire on soldiers they can't see. You'll want these soldiers to have high TU, Reflex, and Strength. If you can get one with high HP that's really good too, but I put priority on the aforementioned stats first before worrying about HP.

Sniper (Precision Rifle + 10KG Armor): Snipers are really good at taking enemies out from afar, although proper positioning is more crucial for the sniper than any other soldier (as far away from aliens as you can get without losing line of sight). Once you unlock Buzzard armor Snipers really start to shine, since you can fly them to the top of buildings where they can get line of shot to large areas of the map while being relatively safe from Alien soldiers. It's a good idea to make sure a Sniper has a medkit on them as well, since if they do get hit they'll usually be too far away from the rest of your squad to have someone else patch them up. You'll want these soldiers to have high TU (since you'll likely be repositioning your snipers a lot), Reflex, and Accuracy.

Heavy Weapons (Machine Gun + 10KG Armor): Very few aliens can survive an attack from a machine gun, since the damage output on them is extremely high. The downside, of course, is that Machine Guns take penalty to accuracy when moving and firing in the same turn. Due to this positioning is just as important on your heavy weapons soldiers as it is for your snipers. You'll want a location with good line of fire, since you won't want to move these soldiers often. One quirk to be aware of when firing Machine Guns is that the stat used for determining chance to hit is the lower of Accuracy and Strength, which means you'll want these soldiers to have high Strength and Accuracy.

Riflemen (Rifles + Any Armor): Riflemen are your bread and butter, being really effective at both medium range and close range (due to burst fire). I like to make sure at least a few of these soldiers have medkits. Choice of armor is personal preference. Wolf Armor makes them a lot tougher, but Buzzard armor allows them to fly around and carry more equipment. For a Riflemen all the stats are important, but I put highest priority on Accuracy, TU, Strength, and Reflex.

Late Game Notes: Once you unlock Predator Armor you'll want to swap your Heavy Weapons soldier over to it. It boosts their strength to 100, and allows them to move and fire a machine gun without suffering a penalty to accuracy. The downside is they lose the ability to walk up stairs, throw grenades, or wield weapons which aren't Machine Guns or the Singularity Canon.

Sentinel Armor will completely surpass all other armor once you unlock it (except for Predator), having higher resistance than Wolf Armor but having the same weight and ability to fly as Buzzard Armor. Due to this you'll want to switch all your soldiers over to Sentinel Armor (except the heavy weapons) once you unlock it.

Squad Composition

For 8-Man squads, I run 2x Shield, 2x Assault, 2x Riflemen, 1x Sniper, and 1x Heavy Weapons. This composition gives you a really versatile squad that can handle any encounter. For 10-Man squads I add two additional Riflemen, since they're they the most versatile class. For 12-Man Squads I'll add two Heavy Weapons users, since by the time I unlock 12-Man Squads I'll usually have Predator armor unlocked, so I'll devote the two extra spots to heavy weapons guys.

Shrike vs Valkyrie

The last thing I need to discuss is your choice of transport ship, in it'll effect the size of your squad. The two transports you'll need to decide between in late game are: The D-42 Shrike and the D-54 Valkyrie. The Valkyrie can bring 12 Soldiers into combat, but they won't start off in a ship, which can leave them vulnerable to alien fire if they spawn near aliens. The Shrike only allows you to bring 10 soldiers into combat, but they'll start off inside the ship, so they won't be as vulnerable. Due to this neither option is really superior to the other in all situations. I prefer to run the Shrike for ground missions, and the Valkyrie for base assaults and the final mission (since you spawn without a ship in those missions anyways).

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.