In all gameplay videos, staves have been the only way that magic has been used. Has anyone seen anything different?

I can't say for certain, but I daresay that you can use magic with your hands as well as with staves. This is an Elder Scrolls game, after all.

I do not believe so, I was reading one of the new articles that were released on the 19th and someone had said that they were playing an Orc Sorcerer and they had picked up an axe and due to that it gave them new skills some melee and some "spells". So I can assume depending on weapon and class you will get at least one spell but do the staffs themselves cast magic when used, I have no idea.

They've said that you can still cast spells with a melee weapon equipped (In some of the recent interviews)