how are you supposed to know in the listening whether it's interesuje or interesuję
Only from context. If there is no context a speaker would add on/ona/ono or make ę audible. With duolingo listening exercises, every time a sentence like that comes up please report it , as it is impossible to hear the difference.
What case is 'chemia' in? Could anybody please tell me how to say 'I am interested in psychology and linguistics'
Instrumental. If you're interested in something, that's Instrumental.
Interesuję się psychologią i lingwistyką/językoznawstwem.
Don't forget that there are no such things as silent "p" in Polish ;)