No, this is currently not possible.
When performing a search, you have four sort options for the search results:
- Relevance: sort by relevancy of the search term e.g. where it appears most times.
- Newest: sort by the date where the question was posted.
- Votes: sort by number of votes. (Think it should be score but not sure at the moment)
- Active: sort by activity date i.e. date of last activity in the question.
What you ask for is to add fifth sort option by popularity or amount of links, which is currently not available for search results.
As for the faq tab of the per site Questions page, the site simply "remembers" the last tab you visited and when browsing questions of specific tag it will still be under that tab. This is not a search, this is the whole questions list sorted by specific criteria.