They use Paypal. The money you earn stays in your PW account and you can use it to bid on ad space elsewhere, but if/when you want to withdraw it, you can do so through Paypal. You have to withdraw a minimum of $10, and any time you withdraw, Paypal takes $1 (even if you withdraw $200! so it's not so bad a deal and I recommend letting your money sit there until you really need it).
I also recommend PW. It doesn't bring in as much as some of the more traditional networks, but the ads are much better-targeted, you have fine-grained control (if you don't like a certain ad or a certain website, you can forbid them from advertising on your site), and you basically set it up and forget about it. The tracking is pretty good, it keeps track of every transaction (ads are billed every half an hour), every ad bid, bid cancellations, etc. It also keeps track of impressions (views) and clicks, but the impressions count seems to be exaggerated and meant more for potential advertisers than for you.
PayPal is as safe as things can be online, and I hear of people getting screwed over by PayPal's policies more often than by their security. If you're using PayPal just for PW stuff, you're not going to run afoul of those policies though.
Now, as for how PW conforms to your criteria:
1. You don't have to pay anything to use PW. I don't even think you need a Paypal account to start, I think that's only for when you want to withdraw the money to use outside of PW.
The ads are not pay-per-click, they pay for the time they spend advertising on your site.
2. You can opt to not allow ads for NSFW sites on your site, but other than that there isn't really filtering. You can also opt to approve every single advertiser by hand if that strikes your fancy (and at the start, you won't have many). However, PW screens all the advertisers, so you're not going to get condom ads or anything like that. Also, although there isn't a genre filter for advertisers, they tend to self-filter since it benefits them to bid mostly on similar sites. I've never had completely left-field ads on my PW box. Many of the ads were for webcomics, although most of them were of different genres. Not really a big issue, I think.
3. There are no popups, and you can choose whether you display "active content" (meaning flash-based ads) and animated ads. Although I've opted out of animated and active ads on my box, from what I've seen elsewhere, even those are usually pretty nice and not disruptive. PW doesn't seem to allow ads that expand to overlap content or start blabbing at you.
4. I covered that near the top of this post.
Some other notes: PW uses an auction system, the highest bidder pays his bid and is the one that's shown in the ad box. There's no rotation, so the ad will only change when it's outbid, runs out of funds, or gets cancelled/paused.
PW has a feature where advertisers can run campaigns that automatically search for suitable sites and bid on them. Although you might get some of your bids from your site's visitors (especially as your comic grows), most of your bids will come from campaigns, so make sure you keep your ad box included in campaigns! The downside is that the campaign bids tend to be more off-topic than individual direct bids. Fortunately, even campaigns tend to filter on keywords, so if you're not overzealous with the keywords on your ad box, you should be fine.
You can always forbid an ad that's currently showing, even if you choose not to screen every advertsier.