1. Any gifts that are 'Dutch' you can pick up at the tax free shops at the airport.
They will not cost you more as in the center of Amsterdam.
2. The train and tram system does work, however with current temperatures might cause some delay.
3. There is a Sheraton at the airport, if you wish to take a hotel in the city, there are several options.
4. Check with the airline on how they handle your luggage. They could send it directly, but with an overnight stop you would have to verify.
5. The usual advice is to be at the check-in two hours before departure. With the recent extra measures I would advide to make it at least 3 hours to be safe.
Once the check-in is completed, go through customs at once, as there usually can be quite a waiting line. Then there is time to visits the shops, get a cup of coffee and run to your exit. Schiphol is huge, so it can take up to 10 minutes to reach your plane.
I never lived in Amsterdam (it was The Hague) but have relocated a bit more south since last year.