Is there any way I can pay for games using my Amazon account? If not are there any plans for such a service?

No, I don't believe there is. Not sure about that one. I haven't read that they are going to implement it, but that doesn't really mean anything either way.

And welcome to GOG!

GOG had payment questions in their survey and some of their decisions have come straight from the data they've gathered from their surveys, so maybe they're considering other payments, just not sure if Amazon wallet is included.

The survey is still up so fill it out, so do it and put in a suggestion for Amazon payment in the comments at the bottom. You get a free game after too!

http://www.gog.com/news/gogcom_annual_survey_get_a_game_for_free