In the comics, Stryker frames Luke by planting drugs in his apartment. So I think that works here too, so they didn't really feel the need to explain that!

I feel like thy left key chunks of back story very vague. Were the exact circumstances of Luke being sent to prison ever explained? What crime was he framed for?
They give hints that it was probably something pretty serious. When Shades, Comanche and Rackham are watching Luke in the courtyard, Rackham comments that with the number of years Carl / Luke is going to serve, he must have pissed someone off greatly. Whatever Stryker framed Luke up for, probably was something as major as drugs or murder.
And what about when he supposedly "abandoned" Stryker in the gutter? Was that before or after he became a cop? Did we ever get any elaboration on the circumstances of that one?
Stryker elaborates on the circumstances when he monologues to, then kills Damon Boone. As he puts it, Stryker and Carl were thick as thieves when they were children, but then they committed car theft as teens. Stryker ended up in juvie, and he was forced to kill in self-defense and ended up in prison. His mother died from cancer after he got arrested, since his father stopped giving them money. Carl on the other hand, managed to avoid jail by enlisting in the Marines. After his military service, he became a sheriff's deputy, which he did until Stryker framed him up for crimes to put him in Seagate.