I've already figured out that Caesar's Legion will attack you on site if Boone is in your party (although I've yet to check if it his beret that they react to or just him), but there are two other things I wish to know:
Do your companions suffer from radiation?
If I give a companion a new weapon, does that weapon require ammo (will they even equip the weapon for that matter)?

no to rads i think maybe on hardcoe?
yes to ammo, only if the weapon you give them is better than what they already have

They only use the weapon if its better then there own. Or if you have them set to mele and you give them a mini nuke, they will still use the mele weapon. However i have seen my companion who uses mele use grenades a few times. They dont suffer from rad poisoning i belive as i just got done with a heavily radiated area and she was fine.