You can check this tutorial about AJAX and this related topic.
Anyway, mobile or desktop, as long as you're hosting your application on some domain and making AJAX's request to this domain, it should work.
To get back datas in JSON, you have to set your server-side script to return in JSON.
For the 'GET' question, I agree I'd like 'POST' requests too. They might be added in a future release.
Be sure to check the How do I FAQ to check more topics about AJAX and other subjects.