Why The Name?
Six Cylinder gets their name from the cocktail that shares the same name, it's made with a cherry brandy and Shirley has cherry in their head! and the six is of course..six.
How did they meet?
Six Cylinder is actually a very very slow burn. They begin with Shirley, who gets eliminated soon after BFB split's and they still manage to end up in the EXIT. From there they're left to "wander" on their own, often getting caught and thrown back into the classroom with Roboty to keep them company. During their wandering in the EXIT they end up opening the door to the Equation Playground where they briefly meet Six before getting caught by Four. A good couple of years pass until they escape Four's EXIT with the other exitors and are met with this looming questioning about the Integer they met.
To answer this burning desire they end up going back into Four with Purple face and literally wander for another year until they eventually find Six. (this timeline differs slightly from the current TPOT)
How'd they get together?
Shirley ends up making a deal with One that deletes them from the timeline and Six makes it their mission to bring them back in any way they can. They never really acknowledge their relationship with Shirley, and they don't want to label what they have out of fear of losing them for being too close. Even after getting Shirley back they're relationship is questionable at best, and a situationship at worst. They'll eventually get comfortable but that day doesn't come till god knows when. they're acquaintances with benefits.
What about the kids?
Bong comes first, during the time Six spends with Shirley, the thoughts of being in a relationship get to their head really bad to the point they get a MASSIVE headache that leads to Four pulling Bong out of their head. Six Cylinder don't confirm their relationship, this is just a kid that they have now that they didn't plan and don't know where it came from other than Six's head. Bong appearing is what kind of start's the whole deal with One. Then there's Painty. Much later, Bong gets a deal with One as well (wow a pattern) that wipes them from the timeline and wipes Six Cylinder's memories of them. One of Shirley's friends ends up trying to bring them back organically by reintroducing Six Cylinder and speedrunning their situationship in hopes they'll just remake Bong but instead they make..Painty. They do eventually get Bong back and they do get to keep Painty.