Suppose I have a proxy IP address, but no infos about it.
Is this possible to know from scratch which kind of proxy it is (distant webservers doesn't know I'm behind a proxy, distant webservers know I'm behind a proxy but doesn't know my IP, distant webservers know both infos)?
Is there a simple test to verify which infos are known about me when I'm navigating?
[edit]
If someone has a simple answer like "no because [etc]", he's welcomed. My first question is "is it feasable".
Lagado Proxy Text mentioned by Joshua Drake in comments works, so it is feasible. The important thing is now to learn how does it work.
Its operation is pretty simple: just be a website. So if you can afford to set up a website which will do exactly the same and will cope with all the traffic you're going to send to it, the problem is solved.
What should such a website do? Check for headers sent by client. The relevant ones are mentioned on the Lagado's page: Via, Forwarded, X-Forwarded-For and Client-ip.
The rest should be easy.
And answering your second question: see BrowserSpy.