1) enjoyable 2) exciting 3) joyous, exciting 4) happy, pleased
3) 흥겹다 is usually used for traditional stuff like old narrative songs, party, entertainment etc. we can say "This traditional Samulnori(사물놀이) is 흥겹다" which may make you wanna dance, or move your body, sing a song because you feel....sorta entertained(?).. um..
FYI, we often say "재미있다(재밌다)"
e.g,
Was it fun today? 오늘 재미있었어?
Did you have fun yesterday? 어제 재미있었어?
This looks fun! 이거 재밌어 보인다!