... we played a few games last night. I played the orks, and got a card that read "as long as you control one or more damaged unit, this unit gains [one power]. My question is:

After battle damage is assigned, do units keep their damage tokens? Or do they automatically heal all damage after my turn is over?

and:

Does the ork unit gain one power whilst in combat, thus dealing one more damage in that particular round only? We discussed this at length but couldn't find an answer in either the FAQ or the rules.

Assigned battle damage remains on unit / capital (unless they are healed by card effects e.g. Archmage of Saphery, Keystone Forge)

Regarding your second question:

Squig Herder reads: Squig Herder gains [P] while you control at least one damaged unit.

This is an example of Constant Effect (pg. 15 in rulebook), which means that the ability will remained active (not only in your turn nor the turn where this ability becomes active) as long as you control at least one damaged unit:

"Card effects that do not have a bold trigger are considered constant effects. Constant effects continually affect the game state, as long as the card is in play and any other specified conditions are met.

Example: Kris has played the Boar Boyz into his battlefield. The Boar Boyz card has the text: “This unit gains [PP] while you control at least 1 damaged unit.” This text is active as long as the Boar Boyz are in play, and whenever Kris controls at least one damaged unit, the Boar Boyz card gains 2 power icons."

Ah - thanks! This makes the game a LOT harder than we played it... Instead of just throwing your strongest units out on the board and grind your opponent, this makes for much more tactical thinking!

Thanks a lot!

The damage works as you put the damage next to the units/capital sections you are dealing damage (one player at a time) to and then apply it. So the damage that would be needed to trigger the squig herder effect isn't on a unit until after damage is dealt.