This sentence is confusing to me. You wouldn't say "does the room have a bathroom?" in English, because either a room is a bathroom or it isn't. I wouldn't say of my bathroom "this is a room that has a bathroom"; I'd say "this is a bathroom".
Would "Bad" more likely refer to a physical bath in this sentence? As in, specifying whether a restroom actually has a bath in it or not?
The primary correct translation here should be: "Does the room have a bath?" Any other translation would just serve to confuse learners, because it is used specific to a context (like a hotel), usually in the sense of a 'pars pro toto', where the bath represents the bathroom.
"der Bad" doesn't exist.
Please don't use Google Translate. It's incredibly unreliable. Use a proper dictionary instead.
nt means the word is neuter.