Translation:He plans to go to the gym in the evening.
what is the difference between 'у фітнес-залі' and 'у фітнес-зал'? When do you use one or the other?
'у фітнес-залі' is used in the sense 'to be in a(the) gym', while 'у фітнес-зал' is used to say that you are going TO the gym. I think somewhere in the course there should be a description of questions 'де?' and 'куди?'. 'у фітнес-залі' is the answer to the former, and 'у фітнес-зал' is the answer to the latter.
Would a Ukrainian be strict about the verb of motion if they're "driving" to the gym? Would some just say іти when it should be їхати?
In this case, the differentiation will not be strict, because the phrase "ходити у фітнес-зал" means to visit a gym on regular basis. Independently of how one actually gets there.
I wouldn't say it's wrong. Just that in English we can call pretty much anytime after dark "night". In Ukrainian night that part of the night when everyone should be sleeping that we might call morning in English (approximately between midnight and dawn) . Right now only the literal translation of evening is accepted, report your answer as correct and hope it gets added to the accepted answers soon.