I've been waffling on this since this past weekend. I'm pretty sure I contributed to a pile-up at the start of a race this weekend. I was in the 3rd row and saw a gap open, went for it and then felt a guys handlebars hit my rear end (nothing crazy either, just pressure). I honestly don't think I was riding over aggressively and don't believe I moved off my line, but wasn't really paying attention to that.
I haven't heard of anyone getting seriously hurt or any major bike damage but it still makes me feel like ill.
Yes. In what situation would you not? Even if it wasn't entirely your fault it's still a nice gesture. And if it wasn't your fault the other guy will let you know that.
And, it's always hard to know what happened. In a stressful situation where there are a lot of moving parts no one person has the "correct" view. Just reach out to the guy and apologize.
It's racing. There are times where it's fine to be agressive and times when you should hold back. Don't dive into a corner and push a guy off his line if it's dangerous. Pass him after the turn. If you can't pass him after the turn, you didn't deserve to pass him in the corner.
At the start, shoot for the gap. Don't chop too aggressively, but the first corner can mean a lot in CX. I wouldn't feel obligated to apologize in this situation, but I'd probably apologize if the opportunity presented itself. Then again, I'm Canadian.
In short, don't worry too much about it.