1.Why are the Chasidim called Stoliners rather than Karliners?
Not all Chasidim are called Stoliners.
2.Why is the name sometimes reversed as Stolin-Karlin?
Because this chasidus, derech avodah, was adopted, effected by the Rebbes and communities of these two towns, villages. At one time here another time there. It actually started in Karlin, then the center switched a few times. (In America, they’re generally called Karlin-Stolin and Stolin for short, in Eretz Yisroel they’re generally called Stolin-Karlin, Karlin for short)
3.Are they somehow more affiliated with Pinsk-Karlin
Actually the reverse, that community is a break-away from the main (discussion not suitable for the hallowed confines of the YWCR)
4.Why do they use the Litvish havara rather than the Chasidish havara?
As was pointed out these towns were in ‘Lita’. There is no real Chasidish havara, chasidic communities of the Polish / Galician / Hungarian regions speak with their local accent. Same with the chasidic towns of Lita / Russia. (Skver, for instance once spoke with a Russian havara) I don’t think the Baal Shem Tov or the Maggid spoke with a Galician havara.
5.Is it correct that they are one of the four original Chasiduses
That may be the case. What is interesting to note is that in earlier times, Chasidim as a whole were generally known as or called Karliners. ???????. That may either be because they were from the very first disciples, one of the first to forge a derech in avodas HaShem, or because of it’s great zest, fervor, fiery approach to Torah and Tefillah. (they were literally the loudest!)