If a character with the Two-Weapon fighting uses an unwieldly wepon in one hand and a balanced one in the other, can he still make two attacks? Can he make a Lightning attack with the balanced weapon for extra hits and still hit once with the unwieldly one? Finally, can he still use unwieldly weapons to hit an engaged target moving away who did not disengage?

In addition, I find the power axe being unbalanced (and Unbalanced weapons in general) and not being able to benefit from the multi-hit talents quite odd. It's not like axes haxe such a better statline that they need balancing. And it makes the fluff weird: apparently Kharn the Betrayer and all the axe-swining whirlwinds of gore of his legion put of less attacks than your average penal legionnaire with a knife and a laspistol....

As for your first question: I would say you could make a single attack with the Unwieldy weapon since I could not find anything saying that you could not or that Unwieldy weapons act differently in Two-Weapon Fighting.

As for your second: Unbalanced Weapons can still use Swift Attack. In fact, by RAW even Unwieldy weapons can be used to perform Swift Attacks. Only Lightning Attack is barred for Unbalanced and Unwieldy weapons.

SephyrBR said:

As said above, unbalanced weapons still get to make swift attacks, and both Lightning and Swift attacks hit a maximum number of times equal to their WS, so Khârn with his 70+ WS will swing more than a penal legionaire with 30WS, since Khârn throws those 7 blows even if most of them miss.

That and we could make the assumption that Gorechild is much more than just a power ax. Even though in this edition its not much more than a glorified one. If you can check out its abilities in 2nd edition 40k. Can't dig my copy out of were ever its gotten to at the moment but he used to kill squads like they weren't there. Also he could still parry not that ability is even part of thegame anymore but more to the point Karn makes the art of murder a graceful affair.