TL;DR:

For developers building web applications on Firefox OS or Firefox Desktop, supporting payments is easy with Mozilla’s fxpay library. In addition to accepting credit cards, it also supports billing to a cell phone #

Is Firefox Desktop the web browser on pc etc and does that mean ordinary websites can bill to cell phones etc through this?

Firefox Desktop is indeed the web browser on PC.

It sounds to me like you can definitely use fxpay to bill to credit cards on desktop. I'm not sure about the cell phone bill thing, though.

Actually you can [theoretically] put a charge on your phone bill from desktop but there are some open bugs around this and it's not a high priority use case for us. The main case for desktop is credit card processing.

Anything special about Firefox OS or Firefox Desktop apps that prevents the use of companies whose full-time business model is payments processing? Or is the special thing about fxpay the ability to pay with your phone bill?

The main reason Mozilla built a payments service and offered a custom library for developers vs. recommending Stripe (etc.) is because no one supported the direct operator billing we needed in our Firefox OS launch countries with a developer friendly interface.

Damn it, now i really want to see a FFOS powered flip phone on sale in the Nordics.

Also, nothing on Firefox OS would stop developers from using any payment processor they wish to. It's all just web technology running on a mobile device so there are many payment providers to choose from, especially for US and European users.

Well it's open-source for one thing. And the MPL is a pretty accommodating license in that it works equally well alongside proprietary code, the GPL or more permissive MIT/BSD licenses.