Πάρτι is a neutral noun, so you need the corresponding article before it, in the accusative (it's the object in this clause). 'Να' introduces verbs.
Now, if you're asking how to say "I like to party", where 'party' is a verb, it'd be something like "Μου αρέσει να διασκεδάζω" or "Μου αρέσει να παρτάρω!" :D The latter of course is slang, used jokingly.
I don't think so, although I'm only a learner, so a native speaker might want to correct me.
'μου αρέσει να πάρτι' would be 'I like to party', which isn't correct a correct translation here. On top of that, I don't think you can use party as a verb in Greek like you can in English. Perhaps in a highly informal context, but it's not listed as having a verbal use in my dictionary or on Greek Wiktionary.
You can use the word stem to make a verb (in a highly informal context as you said), but then it is conjugated normally. Μου αρέσει να παρτάρω in that case ;). Verbs that use borrowed words are always conjugated, unlike nouns.
Thanks for that Troll, I did wonder how to make verbs out of loan words!
Not every loan word can be turned into a verb (that is used by natives). But usually they end in -άρω. Χακάρω=hack, παρτάρω=party, σκοράρω=score etc. :)