I always saw this as a part of Rust's detached, analytical behavior. He's checking his pulse to get an objective idea of himself and the state he's in.
For instance, for the first scene mentioned (altercation with Marty), I can imagine a few different interpretations:
Rust is willing himself to calm down. Though he didn't show it (at all), the confrontation actually got his heart racing. But Rust is committed to being almost pathologically detached. Being flustered won't do. So he checks his pulse to see when he's back to his normal heart rate.
Rust is curious: Did this tense situation even register? Is his pulse any higher than if he was staring out a window? Rust being Rust, he doesn't actually know if it affected him. So he seeks an objective metric: Is my heart racing?
I actually imagine him finding that he's completely unaffected, and concluding (cynically but correctly) that he really is pretty messed up. It's not that he's committed to being detached; he's incapable of being otherwise.
A mix of the two above: Rust knows he's been affected by the situation, but how much? How much impact did this have on a scale of 60-100 bpm? He's trying to supplement his subjective experience with objective measurement. He's not trying to tamp down the emotional/physiological response as in the first interpretation, nor is he unaware of it, but he is trying to reason about it.
For the bikers's raid, the same three interpretations could work.
For the last scene, I'd say the first interpretation seems more likely. He messed up: he didn't act rationally. In a sense, his body betrayed his mind. So he's trying to find his way back to his level-headed, intellectual self.
Regardless of his reasons for checking his pulse, it's a fantastic "mannerism" to give the character. It seems like a very clinical thing to do in emotionally-charged situations, and so underscores how Rust just isn't like other people.