If you are roleplaying a character that can not use magic, does it make sense in lore for them to be able to enchant or disenchant weapons/armor/clothing? Or can only people who can also do magic, do those sort of things?

I think it would count as using magic since your using a soul trapped in a soul gem to imbue something with magic. I suppose a non magic user would buy enchanted gear.

Theoretically a character who can't use magic at all mightn't even be able to use soul gems to recharge a weapon / staff.

I do remember reading the book about white and black souls and the dangers of black soul gems. If touching a black soul gem with the wrong size soul in it could take a piece of your soul into it just by touching it... I suppose an enchanted weapon might be recharged just by touching a soul gem. I'm not 100% on if using a soul gem to imbue something with magic would require the person to use magic but I would think so.