You can raise carry weight through enchanting.
Yes but it's a give and take situation. To get extra carry weight on your gear you have to give up something else. It's much better to just add stamina if you have the points free because with that you can also run longer and use power attacks more and use the enchant slots for something more useful.
If you use magic reduction gear you don't need a ton of magicka. I can cast the Master leve Firestorm spell for only 59 magicka but I am wearing a lot of gear that reduces the use of magicka for destruction spells. If I double cast a Dremora Lord on that same character it will drop me from 626 magicka to 29 but I have no reduction for Conjuration. Double cast allows them to stay out longer. Single cast it still costs 213 so that reduction gear is extremely helpful in getting off multiple casts or if you don't have a big magicka pool.
The biggest problem I see with using such a multi-faceted character is that for melee you need one set of gear and weapons and for the caster you need another and when you switch to archery you'll need another. It's a lot of gear changing and constant gear changing in this game is not fun. The interface is very clumsy and changing multiple pieces really slows everything to a crawl. It's horrible if you have to do it multiple times during a fight. That's the main reason I pick one class and play it consistently and then start another character using a different set of rules.
THe other problem with using a character that does everything is the limited number of points to put into perks. You will be the master of nothing by the end without sacrificing severely in another area.