Is is possible to use the search function in S3PE (under "Tools"), to search in the name tags column? Because now I have to almost look at each individual name tag to find the XML or ITUN I'm looking for.

I've tried the search function, but it keeps being grayed out. It only works if I put in numbers apparently, which aren't much help to me.

No, search is for searching the content of resources. If you want to look for resource by Tag, sort on that column - turn on sorting (if it's not on) and click the Tag column heading. Once you've got that far, you can filter by ResourceType (most of the Tags only represent one ResourceType).