Hand of the Mage already exists in Pathfinder. Though it's really only useful for classes that can't cast it. Cantrips stay in memory and are castable all day long for spellcasters.
Yes, you did it all correctly.
Personally, I don't recommend using 'charges per day' unless it's something expensive, as unlimited isn't that much more.. usually. But sometimes ya have to fit it into the characters monetary ability. 1st lvl wands are the cheapest thing under the sun. Scrolls and a Handy Haversack work incredibly well also.
I play a lot of mages, but I try NOT to get anywhere near melee range, so I rarely use burning hands myself after level one or two.. but that's just my playstyle.
Yea that's right. Note that you can artificially nerf your own caster level if you want when creating items, to make them cheaper. This is why scrolls, wands, and potions are always considered to be made at the lowest caster level; people who craft things are cheapscates and don't like spending more than they have to. :-)