If I give a weapon with a soul trap enchantment on it, to a follower, and then load my follower up with soul gems, can the follower entrap souls like the player character can? I know this is as simple as just give a weapon to a follower and find out, it's just that I don't a soul trap enchantment discovered, therefore, no weapons with soul trap with which to give.

Yes. I gave Lydia an enchanted soul trap ebony sword, which she uses when she isn't using a staff. Once we get back to Solitude, for example, I'll take the filled soul gems from her and recharge her staves. Unfortunately a follower won't use the soul gems to recharge their own weapons, so you have to do it for them.

You just need to make sure that the weapon you give your follower is better than the original, otherwise your follower won't equip it. Also, because there is still a problem with followers using a better bow, I would stick with giving them a sword, battleaxe, etc. instead.