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.

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.