Definitely yes regarding acquiring them from bosses.
As for elemental types... my feelings are mixed. I *love* the colorful presentation in DMC3 and I think the elemental aspects are really excellent from a presentation point of view - it went a long way toward giving each weapon tons of personality.
I'm not a huge fan of weapons dealing elemental damage/enemies having elemental weaknesses, though. I know that goes back to DMC1, but I'd really prefer to see weapons deal consistent damage - there shouldn't be a feeling that there's a "right" weapon to equip to defeat a given enemy (even if, for reasons having more to do with movesets and attack speed than just the elemental weakness side of things, Cerberus is really great against A&R and A&R is quite good against Cerberus).
They don't need to keep the elemental damage, though. They could keep the stat boost that each weapon granted in DT, and maybe give each of the weapon a move that causes an elemental effect on an enemy, like an ice weapon having a move that freezes the enemy in a block of ice. It could open up for a Wild Stomp-type crazy combo that only effects an enemy in that situation.