Will the spells raise my conjuration and the appropriate skill? Just conjuration? Just the combat skill?

Curious as I contemplate a 2nd character play through.

J

Both skills (conjuration and weapon type) will go up when you use bound weapons, but the damage comes from the weapon skill. Conjuration does not increase the damage, though the skill increases when you fight with summoned weapons.

There is a conjuration perk that increases the damage of bound weapons, but I am not aware of concrete numbers as to how much it increases, is anyone else aware of dmg% gains from that perk ?

I can vouch for gaining both one-hand and conjuration levels from bound weapons, if you find a bound sword tome early game I found you can gain quite a few skillups and levels very quickly

EDIT, just found this on UESP wiki

Bound Sword's damage without Mystic Binding is equivalent to a Steel Sword, and with mystic binding is equivalent to a Daedric Sword.

Bound Battleaxe's damage without Mystic Binding lies between Iron and Steel Battleaxes, and with Mystic Binding is equivalent to a Glass Battleaxe.

Bound Bow's damage without Mystic Binding has the Bow's damage between Ebony and Daedric and arrow damage equivalent to a Daedric (unconfirmed), and with Mystic Binding the Bow's damage is equivalent to a Daedric (Exquisite) and arrows equivalent to Daedric.

I would not bother with bound weapons as an assassin, fighting multiple targets could be a problem if your weapon/weapons suddenly disappear mid fight.

Also, don't the bound weapons give off light that will reveal you when you're sneaking?

They make a noise when you cast them that alerts mobs to your presence unless you have the quiet cast perk.

The bound swords are not Daedric level. I wish this would stop being said. They are either Glass or Ebony, but NOT Daedric. The perk adds approximately 50% damage. The axe is exactly Glass level though perked. The axe is actually rather junky if you plan on doing the companions quest, because you get one that is only one less damage initially and then one that does more at the end.

The bow damage sounds about right, as it's MUCH more powerful than Orc and Supple Ancient. It's strange though that the bow requires such a high Conjuration skill, because if you want to play an "Arcane Archer", you are not going to be able to use it for around 15 levels unless you cheese your Conjuration skill.