no, that is not German, you would have to say "Nächster Montag passt mir gut"
Does this mean that next monday is a good day for me in general or that it is a good day for an appointment or something? Can I also say "nächsten Montag" (without the "am")?
It could mean either. Not sure why you would want to say 'nächsten Montag', it doesn't make sense. 'Nächster Montag' is the only correct way to decline 'nächst' here.
I wrote: 'next Monday is fine with me' and was corrected. Isn't it ok to say so in English?
I think that should be accepted as an appropriate English translation. It is exactly what the sentence means. Duolingo often seems to fall into the grey hole where the best way to translate is not always clear: translating a phrase literally (word for word) to emphasize the grammar and to understand how the German language and mind work, versus translating the intended meaning of the phrase. Translating the intended meaning is the obvious choice for some very common phrases. Think of "Ich habe Hunger" versus "I am hungry"--Duolingo would never (I think!) offer up "I have hunger" as the correct translation into English. But in other sentences the programmers, monitors, or whoever "they" are seem stuck on literal translations even if those are not the best or most common way to express the idea in English.
It's OK as an English phrase, but I'm guessing fine isn't an accepted translation of gut.
If you hover over 'gut' in DL's sample sentence, 'fine' is indeed given as one of the translations for 'gut'.
I believe it is strong inflection since there is no article before Montag.