When I kill an enemy (or a droid or a tower) some gold (and silver) cubes drops on the ground. Can enemies pick those cubes? When I get solar directly to me?
When solar drops on the ground - everybody can pick them. Though it happens only if an enemy is killed by something that is not an Awesomenaut. If enemy gets killed (which means - this deals the final piece of damage enough to kill him) by droid, turret, giant worm, solar boss, or just kills himself (Clunk and Ayla may do that), solar drops on the ground. So, answering your second question, if YOU were the one to deliver that final blow on an enemy Awesomenaut, then solar doesn't drop on the ground - it goes directly to you.
Btw, it is not applied only to Awesomenauts. Notice, that when a droid destroys another droid - gold cube drops on the ground, and when an Awesomenaut destroys a droid - solar goes directly to him. Same with turrets. Same with little creeps - the one who kills it gets health.
Oh, and about the question in your comment under first answer, when an Awesomenaut is killed, all members of opposing team get solar, no matter what killed him. If you killed him, you also get additional solar that drops out of him. If your droid kills him, this additional solar drops on the ground. Note that this team bonus (which is about 30 solar, not sure) is applied only when an enemy Awesomenaut is killed, not a droid.