Staff Fighting

Can I use spells while wielding a staff?

Yes, you can, at least when using a magic staff. Haven't tried with regular ones, but I assume you still can.

Hmm. Then I might have to give a thought if I need it. The only place I want it is to beat people I do not want to kill such as the arena, without the need to switch between fireball/sword.

When fighting in an arena, there is no need to actually activate fireball at all. Just place it on the hotbar along with the sword and use sword to win arena fights and fireball elsewhere. That's what I did at least.

With a magical staff its possible.