Assuming my router at home is compromised, what can an attacker see or do in case I use tails or the Tor browser for surfing?
If an attacker owns your router, they can connect to any port on any host inside your network. Any file sharing, screen sharing, other services that are normally not open from the internet, are now exposed to your attacker.
Using Tor does not protect against that.
However, owning your router does not allow an attacker to decrypt/modify your Tor sessions. (Assuming that they do not own the private keys to the relays in your circuits.)
Just like your ISP they will be able to do correlation attack: How much data did you send to the TOR-network? Is there a place in the TOR-network where a similar amount of data exited the network?
The TOR bomb threat failure is a situation where knowledge of you using TOR at a given time is enough to break your anonymity: https://www.theverge.com/2013/12/18/5224130/fbi-agents-tracked-harvard-bomb-threats-across-tor