Is it viable to use the Bound Battleaxe through the entire game or should I go back to using my greatswords? I think I remember reading some where that it does around steel weapon damage, if that's true, that will be a rather low damage at later levels.
I am also aware that there is a perk to increase their damage and that they use my two-handed skills/perks.

I was under the impression, from some source I can't recall, that they did damage on the level of Ebony or Daedric, without any upgrades. I'm not sure, but I don't think it's Steel-grade.

I used the sword with mystic binding for quite some time, I think with the perk it's like daedric.

http://elderscrolls.wikia.com/wiki/Bound_%28Item%29

I'm pretty sure that I read some where on a wiki that the bound battleaxe did steel weapon damage, bound sword could do between ebony and daedric, and bound bow could do the most at full daedric damage, around 90.

According to http://www.uesp.net/wiki/Skyrim:Bound_Weapon, a bound battleaxe does 17 damage, rising to 22 with mystic binding. For comparison, a steel battleaxe does 18 damage, and a glass battleaxe does 22. A daedric battleaxe does 25, so it's ahead of the bound version even before considering that you can smith and enchant it. (All these numbers are the base damage, of course. Actual damage will be higher depending on skill and perks.)

For some reason the bound bow is much better. At 18 damage, it's equivalent to a daedric bow, and the arrows (at 24 damage) are equivalent to daedric arrows. With mystic binding, the bow does 24 damage, between an exquisite and a flawless daedric bow.