A simple question but after 2 hours of google search I still don't have an answer, are companion affected by the strenght requirement on weapon or can they use any weapon regardless of their str score?

Pretty sure it affects them.

Veronica will choose to wear Pushy over the Ballistic Fist because she doesn't meet the latter's STR requirement. I'm not sure if she actually punches slower with the Ballistic fist, but that's worth noting. The Ballistic fist normally has better damage all-around, so if she opts for Pushy, it means the game is actually calculating out her DPS with both weapons and applying the slower punches she would have with insufficient STR. I doubt it would bother calculating this for her if it has no effect at all.

I HAVE given Boone an Anti-Materiel Rifle and then taken it away after noticing him missing fairly often, and he performs far better with a Sniper rifle.

Yes companions are affected by the strength requirement of a weapon, thats why the ballistic fist loses out to pushy for veronica, companions and every npc have as special score and it is used for multiple things like determining wether an enemy scores a crit.

Yes.