You are right about "in" and "of". "with" is less common in my experience and I would say that it is sometimes interchangeable with "in", but never with "of".
For me, "with" is used with a method or a phrase which describes the use or application of something e.g. I have experience / I am experienced with/in the use of laser technology," (I am not really!) I prefer "in" to "with", even when "with" is possible.
On your second point, consider this situation:
"I went on holiday and tried 10 different sports including skiing, on 10 different occasions. I had some amazing experiences.
Are you now experienced in skiing (the skill)?
No, I have very little experience of skiing (the activity). I have only had one experience (occasion) of skiing."