- does a fullnode after a power-outage (e.g. 24h) automatically re-sync?
Depends. For Bitcoin Core, if your client gets shut down unexpectedly, your database might get corrupted. If that happens, you'll have to reindex the whole database which would take quite a while.
- Does such re-sync entirely depend on the speed of my internet connection?
No. It's mainly dependent on your processor and also the speed of your hard disk. It's the same with when your client gets synchronized the first time.