Only local traffic, it's quite simple. Unfortunately, no.
DB likes things complicated. For example, some regional trains are operated by DB Regio, but others by DB Fernverkehr. On the former, the 9-euro tick applies, but not on the latter. Unfortunately, these trains are not clearly marked. To do so, you have to open up the additional information in the Bahn app and read the fine print, so to speak. According to Deutsche Bahn, there is no way to prevent these trains from being displayed in the search.
Regarding the HVV semester ticket: This is valid as a 9-euro ticket throughout Germany on regional and local transport. Clearing of the refund of the difference between the amount already paid and the 9-euro ticket is still in progress.