Not necessarily, you don't wait until you feel your body weight drop during a choke to tap nor should you wait to tap until you feel the onset of soft tissue damage. Listening to your body is important in all martial arts training.
Exactly, if the tap is late, meaning they could of tapped sooner to prevent injury. Knowing when to tap is very important which is overlooked in some BJJ schools.
That's very uncommon in BJJ, fringe injuries occur in all disciplines. I had a friend tear and dislocate his shoulder in Aikido, but that's far from common in Aikido.
I've had someone try to check my leg kicks incorrectly and they injured themselves in the process. Meaning sometimes people put themselves in horrible positions that will inherently make sparring more dangerous for themselves. However, that doesn't make the training itself dangerous, it's the training partner making the situation needlessly more dangerous.