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I am using bound sword spell as my main weapon. If I get one handed perks like increase damage or reduce stamina etc will it effect the bound sword as well? Its attack damage will also increase?

Thanks

I would think so, but haven't tried it myself. The wiki says

•Using bound weapons in combat also increases your Conjuration skill, on top of the skill associated with the bound weapon used (One-handed for the Bound Sword, Two-handed for the Bound Battleaxe and Archery for the Bound Bow).

So it's taking into consideration your one-handed when you use the bound sword.

Yes on all counts - think of the bound/mystic weapon (depends on perks) as equivalent to a steel/daedric quality weapon which is zero weight which you summon rather than carry around with you. They level both weapon skill and conjuration, except the bow which only levels archery.

Long term, they aren't very good, as tempering and enchanting easily makes normal weapons more powerful, but early game they are very useful.

All three bound weapons are available at level 1, but only sword and axe can be bought - the bow's spell tome is found in a bucket in the first room of Fort Amol Prision, but the spell can't be cast at level 1 without fortify magicka effects, for example, being an altmer wearing the novice hood found in Helgen during the tutorial.