I asked a BJJ coach about running. He told me that running a long distance with the same pace affects the muscles negatively. It makes you react slower during a BJJ fight. Is this true? If yes, how should a BJJ practitioner run?
In Living the Martial Way, Forrest Marshall makes the insight that you should run like you want to fight: if you want to fight at a slow, plodding, constant pace, then run long distances at a slow, plodding, constant pace. This advice is a gross simplification of exercise science but it is basically true in this context.
Your coach is right. Long-distance running is close to useless for BJJ. It isn't very good at improving cardio for BJJ and it interferes with other better forms of training for the sport (like strength, power, mobility, or technique work). If you run at all, it should be short sprints.
Your cardio for BJJ should consist of huge amounts of BJJ. Specifically, lots and lots of long, high-intensity rolls. Shark tanks, where you stay in the middle sparring while fresh opponents come in every few minutes, are a great method of doing this, but good old long-rounded sparring classes are solid too.
Other than sport-specific cardio, you should be thinking about strength, explosiveness, and mobility. Those are developed through strength training programs (probably with barbells or other heavy resistance implements), power training programs (probably with extensive use of sprints, jumps, and Olympic lifts), and yoga or stretching.
I don't think he's right, but he's not 100% wrong either. When you're running, you should be focused on interval training, e.g. jogging for 30 seconds, then sprinting for 20 seconds, then slowing down to a jog, then sprinting. Rinse and repeat until you feel like dying.
Stop thinking so much.
Rolls for building rolling cardio.
Wrestling for building agility during scrambles and standing explosiveness.
Sprints and hills to help with explosiveness.
Long-runs for general aerobic fitness.
Do them all.