PHPbb is probably going to be your best bet, it has a boatload of addons that should provide whatever special functionality you need. I'm having a hard time even thinking of another forum package that's been actively developed since 2005 or so, excepting WordPress--which isn't really a forum--and large companies' in-house solutions integrated with all their other products and services.
edit: Oh, and to address your issues from some simple forums I've set up in the past:
1. I'm not familiar with this, not sure how it would work. Are you trying to paywall the forum itself? SA.com is the only forum I can think of that's successfully done this, personally I wouldn't be looking at them for protips but whatever.
2. It has a robust group-management system built in, I've done it. Not sure quite what you mean by badges, like small icons in addition to the avatar? Should be possible, I've not done it though.
3. This is really easy, I've done it.
4. This is really easy, I've done it. Can be based on permissions if some groups shouldn't be able to do it.
5. Never heard of something not working with google advertising, but you'll likely need to alter some of the PHP files yourself.
6. Voting, like topic polls or upvote/downvote stuff? I've done the former, and can't imagine the latter not having at least a dozen modules.
7. This will require some coding but it doesn't sound difficult and you'd have to do it for any forum technology.
8. It's easy to limit the size of signatures and disable markup. It won't filter objectionable content out of the box but there's probably modules, most likely they will apply to the whole forum though not just sigs.
9. Disable bot access, it's in admin settings somewhere. Can do the same for unregistered members. I've done both, and also nuked the member list though that required some coding. Do be aware that if someone really wants to find a way in, they will...web forums are inherently rather low-security systems.
10. Will probably require coding. But you're using that money to pay the coders, right?
11. Not sure what this is.
General advice: be prepared to deal with whiners, spammers, and jerks. Especially if there's money involved. Setting up effective moderation and user access regulation ahead of time will pay dividends.