Not anymore, they use .se now.

EDIT: This is wrong, see peteri's post.

.org is essentially the same as .com, owned by the US government and managed by Verisign.

.org is operated by PIR (actually afilias in dublin according to wikipedia) not sure exactly how much leverage the US has over it currently.

I imagine that corporate interests (records, movies) in the US and EU could collude to get a pirate site taken down if it had a .org, it would just be another tree to fall.

So far the US has not publicly tried to drop wikileaks.org, forcing WikiLeaks migrate to a TLD that is clearly outside US reach. Maybe they wouldn't want the bad press.

I'm wondering what is possible if I simply have a relationship with a reseller in the US (e.g. the one with the superbowl ads, or eNom), but my TLD registrar is outside of reach. I assume they could make the reseller drop the contract, or terminate it.

No, you're right. Thepiratebay.org redirects to thepiratebay.se