1) Did I do something wrong regarding this incident? On and with don't sound as quite right. An incident is an event and it is hard to be with or on an event.
2) I avoided seeing her . (Which one is correct?) This one
3) I was happy to meet new friends. I was happy meeting new friends. Both are OK, the second one is better
4) I failed to do it/ I failed doing it. Both are OK but mean different things. The first usually means you never even started it. The second means you failed while in the process of doing it.
5) I'm currently on holiday. This one is correct