Can follower take items all on his/her own?

Arrows have a chance of being added to the inventory of an NPC or creature they hit. There is no animation for pulling them out, they just instantly end up in their inventory.

NPCs without a weapon in their inventory will go after weapons laying on the ground and in containers, perhaps that's the reason for melee weapons in your companion's inventory. They might also go for unattended staves, though I've never observed such behavior.