ad 1. yes, the connector authenticates properly. Unless it does not, which would mean there is a bug in it . But let's assume at this point that auithenticatin is fine
ad 2. XML query indeed is not trivial to construct. For saved filters, it is something like:
Where jira-url is the base URL of your JIRA and <filter-id> is a numeric ID of your saved filter (they start from 10000 - you can find your saved filter ID in your saved filter URL - it in the requestId parameter)
The fact that fixVersion is there in the filter query is weird. Are you sure you did not specify fix version in the fiter definition in the connector? Note that fix version is one the "Components/Versions" tab (not selected by default). I find it a bit puzzling why this could happen. Please see the end of this comment for an idea how to debug this problem
ad 4. saved search URL is described in my response to your question #2.
I think it would be a good idea to find out what URLs are queried by the connector by consulting your web server log. If your JIRA is fronted by Apache, the log will be somewhere in the file named access.log (usually /var/log/apache2/access.log). For IIS, I am not sure where access logs are.