This is more on the advertiser side rather than affiliate.
So there's this android apk offer I saw on a few networks, pop birds.
You install it, and to buy some in app purchases you click a button and it charges your carrier bill. no confirmation, no are you sure button. you just get an sms message that you've been charged
my question is: how do they do that? what services do they use for in app carrier billing?
Please point me to any services/resources about the subject.
Not exactly sure but maybe shortcodes?
You mean something like https://fortumo.com/in-app-payments !?
I looked into fortumo and it looks like they take 3-6 months to get paid.
Are there any other similar services you know of? I found text2pay.com but seems like they take a hefty cut (carrier 35-45%+they take 10%).
Yes, I think these games developer will automatically charge through those carrier billing company.
One way that you can find a carrier billing company is through reverse-engineered all the pin submit offer from your cpa network. Open the landing and look at those T&C below the page, you can find the carrier billing company. Google the company name, if you're lucky you can find their website 
For the UK, one of my clients use http://www.payforit.org/
You need to arrange these things for each country, as it is tied to carriers.
Look at the world mobile congress site, under billing providers, you should find many countries represented.