1. The bag will be checked in through to your final destination. You do not need to contact the EU customs (at HEL), unless you import goods.
2. Yes, you will have to go through Schengen area passport checks at HEL, after the transit security check.
3. 3 hours is not enough to get to the city center and back. Even if the ring train line were ready, it would still not be enough. Check the Finavia website for airport facilities. You can also visit the airport vantage point in the Hilton hotel (right next to terminal 2).