Yes there is a small safe, big enough to leave travel money, passports and tablet/notebook inside. You close the safe with whatever numbers you want to use as security and open again with same numbers. Instructions are on the safe.
Yes! It is about a 10 minute walk or so from the old town square. There are also a lot of bars and restaurants en route so if you feel like popping in somewhere on the way back to the hotel it is perfect :)
It is a bit further of a walk to the castle but we walked it anyway since you can see a lot on the way (and the weather was great!). There are tram stops right down the street from the hotel you can take up to the castle though.