Well, the icons above the soldiers represent how much damage they can/have taken and how much cover they have versus a given opponent. it also shows how many movements the soldier has left.
So, you see a bunch of smaller dashes. if they are all solid, the solder has full health. If it gets hit for 3 damage, three of the dashes will appear empty. The remainder will remain solid.
Next you have a shield. If it is full, you have full cover versus the opponent. if it has half full, you have some protection. If it is empty, he is out in the wind. and if it is yellow, he is flanked by an opponent.
Finally, you have two larger arrows. Your soldier gets two moves per turn. If you move once, one of them goes away. If he moves twice (or dashes) both go away.
As for Buffs that the aliens have, I know that they can mind merge with each other giving them some bonuses to health and accuracy, but couldn't give you more than that. I am sure other players will speak up and fill you in on that.