Does a toon have to be at a certain level to max out their professions? As an example, if I have a level 20 mage, can i get their Alchemy level to 300? This ,of course, is with a higher level toon supplying the mats.

Yes, there are level requirements to train in expert/artisan/master for I think all of the crafting professions. I think in most of them the highest a level 20 could go would be expert.

Yup - there are level requirements to train each step of a profession. For example, to train Artisan Alchemy, you must be level 30.

I know that I needed to get my enchanter up to 50 before he could learn past the 300 level.