Over the last few days I have built a proof-of-concept demo using the GSS-API and SPNEGO. The aim is to give users single-sign-on access to
To answer your first question, GSS-SPNEGO may include multiple round trips. It is not limited to just two. You should implement a session handling and upon successful authentication issue a session cookie that client should be presenting on each request. When this cookie is invalid, server would force you to re-authenticate. This way you would only incur negotiate costs when that is really needed.
Depending on your application design you can choose different approaches for authentication. In FreeIPA we have been recommending to have a front-end authentication and allow applications to re-use the fact that front-end did authenticate the user. See http://www.freeipa.org/page/Web_App_Authentication for detailed description of different approaches.
I would recommend you to read the link referenced above and also check materials done by my colleague: https://www.adelton.com/ He is author of a number of Apache (and nginx) modules that help to decouple authentication from actual web applications when used with FreeIPA.