Yes, the path setting on the cookie matters - it tells the browser which part of the site the cookie applies to. Set it wrong and you won't get your cookie back!
I mainly use two tools for debugging cookie problem - first, Firefox with the Web Developer addon. This gives you a screen where you can see what cookies are set and all their parameters. Failing that I pull out the big gun - wireshark. It's a network-level protocol examiner and you can use it to see exactly what the client and server are actually saying to each other. The learning curve is steep but sometimes you really need to see what's actually on the wire to solve a problem.