German divides things up a bit differently than English :)
"Burg" is a castle (fortified place). "Schloss" can either be a castle or a palace, but is usually less strong/defensive than a Burg, I think. "Palast" can only be a palace.
I'm not sure whether every "Palast" can be a "Schloss". Certainly not every "Burg" would be a "Schloss".
I guess that the main point of a Palast is luxury; the main point of Schloss is that nobility live there (which sometimes needs a bit of strong masonry to protect them, depending on location); and the main point of Burg is the defence.
So if an army garrison is stationed there, it won't be a Schloss but has to be a Burg, while a similar building housing a local lord instead could be a Schloss.
There's also the French loanword "Palais" but I'm not sure exactly where that fits in.
That would be the pre-reform spelling, so you would be rather out of date.
You may come across that spelling, e.g. in older texts or even in newer ones written by people who grew up with the old spelling, but it doesn't correspond to the orthography currently in force.
I never got an email saying that I would get a reply, but I did have a couple where they said they had made a change. I suspect that a lot of people had reported the same things. And I have reported a lot of other things that I have never had a response on. I think part of the problem is that they will fix it only on the reported exercise, but it will still exist in other exercises.
Yes, Schloss means not only a castle but also a lock as in the lock on a door or a padlock (which is more specifically a Vorhängeschloss).
"to lock" as in the verb is abschließen -- in the simple past tense, it looks a lot more similar to the noun, e.g. Ich schloss mein Auto ab "I locked my car". (Though you'd probably hear the compound past more often: Ich habe mein Auto abgeschlossen.)
It's not a lock as in a lock in a river, though (for ships to go up or down), or an airlock. Those are eine Schleuse (airlock: Luftschleuse).
About "schön". Way back. I learnt schön = beautiful/handsome, hübsch = pretty & nett = nice. I also got tired of typing beautiful and typed "This castle is handsome". I was wrong. I can wear that; but I got "This castle is nice" as the accepted translation. I would never translate schön as nice. Am I wrong and out of date?
"This castle is beautiful" is a good translation and is the default translation.
Additional accepted translations are unfortunately not always completely uniform across the course and may depend on the person who added them and/or on what people had suggested as alternative translations.
The equivalences you learned are good (though I would personally use gutaussehend "handsome" for a man rather than schön "beautiful"), and "nice" is not a very good translation for schön.
Calling it "the accepted translation" is misleading; it's not the only accepted translation nor the one that's considered best. It may be "an accepted translation" as sometimes the course is a bit lenient in what it accepts -- sometimes more, sometimes less.