You can achieve this with a bit of an hack and the special properties bountied/featured questions have.
This search will do the trick in 99% of the cases:
intags:mine closed:no lastactive:7d.. hasnotice:1 locked:no
Let's dissect what is going on here:
- intags:mine returns only posts that appear in tags you have marked as favorite.
- closed:no returns only questions that are not closed
- lastactive:7d.. for posts active in a specified time period
- hasnotice:1 returns only posts with a notice shown below them
- locked:0 returns only posts that are not locked
Why does this work?
Questions that have a bounty on them can't be closed, bounties can only run for 7 days so they need to have been active the last 7 days, and the maybe surprising trick: the bounty text is actually a so called post notice. Post notices can also be added when a post is locked so we rule out those keeping the questions with most likely bounty notices.
Now it is possible to have a question with a post notice, that is not locked and also not a bounty. That is why I said for 99% of the cases. Your search might return a false positive now and then.
For info on the other advanced search terms see the Help topic on searching