berlin-gdansk-budapest trains

Don't know about Berlin-Gdansk, but from Gdansk to Budapest there won't be any direct trains. Your best bet would be to take a train to Warsaw in time for the overnight train to Budapest, and to shell out an extra 15 euros or so per person to travel in a locked, quiet, safe sleeper train. The overnight trains around eastern Europe tend to run about 30-40 euros Warsaw to Budapest is a long trip so I suspect you're looking at 40=45 each way.
Remember that overnight trains tend to be a lot busier on Friday and Sunday night. If you travel any other night of the week, you should be able to simply walk up to the ticket desk and get a sleeper on the spot. Otherwise, get your ticket at least a few days in advance.
Don't plan on sleeping unless at least one person in your compartment will be awake. Overnight trains are a magnet for thieves, although it's pretty well impossible to be robbed in a sleeper car - too few suspects for anyone to try.

Berlin-Gdansk goes for around 40-50€ in a couchette (night train). It's way cheaper to take a train to Szczecin first from Berlin (approx. 10€) and change there for one to Gdansk (day train around 20€ 1st class). Trains are very safe in this part of Europe and it is simply bullshit (sorry for the language) that you should not "plan on sleeping unless at least one person in your compartment will be awake"

Maybe the trains between Gdansk and Berlin are safe, but those running through Warsaw are not. I've been threatened-robbed once on the Warsaw-Gdansk night train and had someone try to steal my bag (caught them in time) another time. Warsaw is the problem - the main station attracts all kinds of types you wouldn't want to meet in a dark alley.

naturally I'm talking about non-couchettes and theives. Theives can't enter couchette cars unless they have a reservation.

Yep, couchettes are ok as is sitting in day trains. Sitting in night trains is a different matter: usually you will be fine, but you are right - Warsaw is a problem, so at least there one should watch everything carefully and fall not asleep if you just booked 2nd or 1st class sitting.