Positions are unlocked by filling the bars in the club report by completing the story, finding the heart icons while a girl is following, the bright pink ones, NOT the ones with the crown on top that are red, and interacting with girls. Interacting is filled by both just talking and having sex with sex filling the bar more than talking. The club report is accessed through the document icon in the club room. You can also unlock positions for each individual girl by leveling the stats you see after sex to 100%. There are also several positions that are exclusive to the Kplug mod. Threesomes are available only if you have the After school DLC, you have this if you use the repack offered here, and you catch girls in lesbian H and peep on them.
No, Party is literally just the game with no DLCs and no mods, but with the official English translation. It does not offer any content the normal Japanese release does not. If you downloaded a repack version or the normal Japanese release, you do not need Party.