Three questions:

1) do power batteries take pushing damage? It seems like the would since they are given actions, but don't have indomitable.

2) if the green ring is assigned to the battery does that give it willpower?

3) if a character like Superboy from SLOSH, Harvey Dent (or any other figure that grants willpower to adjacent figures) is adjacent to the battery would it get willpower?

1: yes

2: The ring is only used by corps members (when assigning it in conjunction with the battery). The power plant is never a corps member because it is not a character assigned a construct.

3:
THE INSPIRATION OF THE 31ST CENTURY: Superboy can use Leadership and Super Strength. Friendly adjacent characters with a lower point value can use Willpower.

I BELIEVE IN HARVEY DENT: Friendly characters adjacent to Harvey Dent at the beginning of the turn can use Willpower until the end of the turn.

The lantern battery is not a character so no.

This had me confused too. I agree that the power battery itself does not get to use willpower if the green ring is attached.

The ring is not "assigned" to corp members. It is assigned to the power battery itself.

The first line of the card states "the power battery must be assigned a power ring..."

However, I think that being "assigned" something does not mean that you get to use the power or ability printed on the card.

Yup the only way to gain access to the rings abilities is to be a Corps Member.

Well normally when you're assigned a relic you get the effects of it. The green ring is most definitely a relic, and the the power battery is definitely assigned the ring.

Now here is where things get murky.

We have never had a resource before that occupies the field and does not generate characters. The Batcave is the closest thing and that made up to 2 characters (though still occupied the field).

According to the rule book it's a character assigned a relic that gets the abilities.

Now we've already have proof that sometimes the rule phrasing is not in line with the intention of new abilities/figures only because nothing like it existed before. Prime example: Horde tokens say they can only be stacked if the started the game friendly to you. With the new Larfleeze from WOL they had to go back and state you can stack the tokens he makes even after te beginning of the game, because when the phrasing of the rule was made there was no way to bring in hordes from outside the game.

On the same note the battery has standard and special powers, would they be counterable?

The Power Battery gives us direction on how the Ring is assigned. The battery is most definitely not a character.

Force Construction - The Power Battery must be assigned a Power Ring (ring symbol) and 1-6 distinct Constructs (construct symbol). The Power Battery costs 2 points, plus the combined cost of the assigned Power Ring and Constructs, and must total a minimum of 16 points. The Power Ring and Constructs are all attached to the Power Battery at the beginning of the game.

The battery is a resource as per The card with it R100.

OUTWIT Give this character a free action to counter a power or a combat ability possessed by a single target opposing character until the beginning of your next turn. Any game effects with a duration specified by the countered power or combat ability
are removed. A character using this power must be within range (minimum range 6) and line of fire to the target.

Again the battery is not a character so you can't outwit any powers on it.