I play a conjuration mage and I purchased the bound sword right from the early beginning and it was at novice level. I'm at level 20 now and its still novice level. Does it change by itself or do I have to find a merchant or mage who sells a Apprentice bound sword or what?

My Frost attronach is already at Adept level.

Spells dont change level.
There is only 1 type of each bound thingy

Spells don't scale. Bound Sword is a novice level spell and will remain novice level. Conjure Frost Atronach is an adept level spell and will remain like that.

Your skill level in any magic school only affects magicka cost. You can however buff up your spells by investing perk points, for example Mystic Binding improves bound weapons.

With the Mystic Binding perk it's the same as a non smithed daedric sword. The only way to increase it's damage is with one handed perks/enchants/potions