How can I detect whether the user accepted my cookie?
Here's a clever trick: use a redirect. Drop a cookie on the HttpServletResponse object, then call response.sendRedirect() to a "phase two" servlet. The "phase two" servlet then tests whether the cookie was sent back to it. If so, that means the user accepted the cookie the first time; if not, it means that either the client rejected the cookie, or the browser doesn't support cookies.
Note that this technique only works if the "phase two" servlet is hidden from the user; if the user can jump directly to the test phase, then the servlet can't tell the difference between newly-arrived clients and cookie-unfriendly clients. That means you should send a redirect from the test phase, to make sure the user doesn't have a chance to bookmark the test phase's URL.