And if I give anergy weapon to them, will they use it and make progress with it ?

Yes, they get new levels, but their skill priorities is hardcoded. For example - Cassidy got small arms and brawling skills, Markus - heavy weapons and such. They got leveled up once every 4 player levels (white "bark" commentary appears).
Additional info on their characteristics and upgrade process can be found in "Fallout bible".

First Fallout system also works this way.

In first Fallout, the only companion that can use energy weapons is Katja, and only can use the plasma pistol and the alien blaster. Note that she not use the laser pistol.

For Ian, the best weapon that he can use, is the .223 pistol, and Tycho, the sniper rifle.

I thought the best weapon for Tycho was the Red Ryder LE BB gun?

Yes, but the ammo for this one is limited. No one have a permanent stock of this amno, and you cannot find it in enemies bodies.

That's true, I found most of my BB ammo early on in the game. Still, I had about 200 rounds by the time I got the gun, and that's lasted for almost the entire game.