is it possible to do the check-in and drop my luggage in the morning? (= what is the expected check-in time for a flight to Europe?)
Possibly, but I wouldn't count on it, either by policy, or just because the check-in counters are closed. For example, if you're flying Norwegian, I believe they only operate a 1-2 flights most days, and their check-in counters won't be staffed in the morning. I would check with your airline to see what time the counters open and how early they can accept bags, a quick message to them on Twitter or Facebook should get you the answer. Their website will also provide their recommendation for how early to arrive and the deadline when check-in counters close. The airport does not have luggage storage or lockers, so you may need to store your bags at your hotel if you can't carry them with you.
is there something to do around OAK if I have 4-5 hours to spare?
Near the airport, not a lot besides visiting In-N-Out Burger. There is the small Oakland Aviation Museum; you can decide if that's somewhere you want to visit based on the reviews online and your interest level. If you're unfamiliar with the area, for safety reasons (that link has sugguestions for other areas you could consider staying too), and to generalize a bit, I would not recommend walking around East Oakland.
If you get to the Coliseum BART station (take a taxi/Uber/Lyft from your hotel, or get to the airport terminal and pay an extra ~$6 plus the regular fare to ride the train to/from the airport station), you can be in downtown San Francisco in 23 minutes for $4.80 (check schedules, but trains run frequently during weekday daytime hours). You can also visit Berkley or downtown Oakland. While you won't have a ton of time, you could easily explore a little or get an early lunch. You could also ride a few stops further into the Mission District of San Francisco, and perhaps get a burrito (though not really the food I'd recommend before a longhaul flight, perhaps Tartine Bakery instead?).