Just dont say kissa
Is 'pussa' incorrect here?
No, that's fine. Both words translate to "kiss", the difference being that pussa is more of a quick smooch or peck while kyssa is an intimate act typically between two lovers.
don't have to add "att" before "kyssa" to make it "I want TO KISS you"?
No, you don't put att after modal verbs.
Is kysser for past tense or something? I was not sure which to use.
The past is kysste, but you use the infinitive kyssa because you already have a verb in the present before it - vill.
Does 'Jag vill hångla dig' work also?
hångla med dig, though that doesn't mean "kiss" but "make out", and it's a little more vulgar-sounding than in English. If you're British, "snog" is a good equivalent.