This is exactly the kind of thing that the NodeBrowser is useful for. If you haven't played with it a little bit, I'd definitely recommend spending some time to familiarise yourself with it. It's found in the Administration Console.
The storeId of a user sandbox is "WebProjectDNSName–UserId", where "WebProjectDNSName" is the DNS name you configured the Web Project with when you created it, and "UserId" is the user id that the user logs in with. So for example if I have a Web Project with a DNS name of "test" and my user id is "pmonks", my user sandbox in that Web Project (assuming I'd been invited to it) would be "test–pmonks".
Your directories will be exactly as you created them, but with two "synthetic" directories added to the front:
"/www/avm_webapps". So in your example, the full directory path would be "/www/avm_webapps/ROOT/products/myproduct" and the full AVM NodeRef (using my example user sandbox above) would be "test–pmonks:/www/avm_webapps/ROOT/products/myproduct".
I'm not sure why the Web Client UI creates these synthetic directories, but they are a minor nuisance when interacting with the AVM via CIFS, FTP and/or the APIs (where they are visible - the Web Client itself hides them). Perhaps one of the engineers can enlighten us?
Cheers,
Peter