Most of it is explained in the movie:
Yusuf: We're too heavily sedated to wake up that way. [...] Can't even think about trying to escape [from Limbo] until the sedation eases.
Cobb (to Saito): When you wake up you won't even remember that we had an arrangement. Limbo is going to become your reality.
So let's tackle the relevant points one at a time.
Why Saito didn't kill himself in Limbo:
So Yusuf's sedative places a person who dies in the dream in Limbo without their conscious minds. This explains why Saito didn't kill himself immediately. However, even when the sedative wears off, enough time has gone by (in accelerated dream time) that a person can't tell apart Limbo and reality after spending decades there and will no longer know that they have to die there.
Why Saito asked "Have you come to kill me":
There are two possible explanations.
This doesn't necessarily mean that Saito knows he has to die to leave Limbo. In the Limbo world, Saito runs a (possibly organized crime) empire just like the dream in the beginning of the film. When a stranger washes up on shore with a gun, Saito might simply suspect he's an assassin.
Saito is confused about what's real and what's not, just like Mal was. He vaguely suspects that someone will try to kill him to get him out, but he isn't sure it's the right choice until he sees Cobb and they remember their previous conversation together. It's even possible that by saying that, Saito actually reminded Cobb what he's there to do.
How it relates to what happened to Cobb and Mal in Limbo:
Cobb and Mal willingly entered Limbo together, and Mal was eventually reluctant to leave it. Cobb had to persuade her otherwise. He succeeded and they committed suicide together in Limbo, but playing with her subconscious like that also had devastating results later in real life.
A somewhat similar thing happened with Saito. Having arrived in Limbo with no memory, he stayed there for decades. Cobb, having entered Limbo voluntarily, eventually found him there but was notably already in a confused state. Saito and Cobb had to remind each other that they're in Limbo, which was probably what helped Cobb and Mal escape in the first place: You need another person to remind you who and where you really are.