In midgard every dual wield class always swing with both weapons. So training left axe will give you just styles, or also it increase your weaponskill and/or dmg?

they allways swing both hands? o_O

Left Axe works different to Dual Wield/Celtic dual. While the latter 2 are similar (beside different styles).

With DW/CD your spec level determines your style damage (maybe without the buggy level 50 style) and the chance your offhand will swing.
With Left Axe your Offhand will allways swing but the damage of both, your main and you offand weapon will be reduced, depending of your LA specc (higher LA, the less the reduction is).

The interesting thing is, that when your specc level in LA and DW/CD is equal you will do the same base damage over time. So they do work different but give the same results in that (unlikly) scenario. One may outdamage the other in a short time, depending how lucky the DW/CD user is with his offhand swings, as LA hits constant (but lower). But over time it will be equal.

This wont be true if you hit styled or have damage add. Here LA is superior because his offhand will swing everytime and the damage add does not get reduced.
Furthermore the haste effect of a fast offhand weapon leads to better results for LA when hit styles.

At least that is what i recalled from my live times and may be pre LA nerv. Correct me if I am wrong.

So LA specc will improve your over all damage output and give you styles similar to DW/CD specc.

Thanks aladrain, your answer seems to verify all my mini test^

the difference is very reduced at high level, cause anyway , even with CD/DW you'll hit 2/3 with 2 wapons.

the difference can be with assassin classes and the poisons uses, on midgard you are sure your both poisons will be used at first hit.