I was boosting my Block skill just now with Spellbreaker and I started to think, if I were to block spells instead of arrows or whatever else would my Block level up or would my Alteration? (I think Ward spells are Alteration) Does anyone know what would level if I were to block spells with my Spellbreaker?

This is just a guess, but I'd say either enchanting or no skill would increase. I say enchanting because I can't recall right now if using enchanted items increases your enchanting skill at all in Skyrim or not.

They do not. Recharging weapons increases your Enchanting slightly.

I don't think that shield would level alteration but I could be wrong. Wards are in the restoration school so if any it'll level there.

According to wiki, it does not contribute to restoration skills. If you think about it, it's fair system as there is no penalty to magicka.

Wards are restoration, but the ward provided by the spellbreaker is only magical (not physical). So it provides no extra protection against melee or archery.

I don't even think wards increase your restoration skill.