It is my understanding that in some game worlds you can establish two holdings with the same user. These two airlines will not be able to cooperate (only be a member of the same alliance) in order to not make an unfair competition. But it is possible for one of the holdings to buy a subsidiary from the other holding and then interline as usual?
E.g.: Holding A has the subsidiary A1. Holding B has the subsidiary B1. May then holding A buy the subsidiary B1 and start interlining as usual?
I wish I could do just that. But I don't think you can do any transaction between two different holdings of the same user - including between their subsidiaries.
Should you desperately want to do IL with subs of different holdings owned by same user, you will need the help of a third party. But then again, if someone found out about it and report it, you will have to terminate IL or if it a recidivist, then other consequence could also be considered.
It’s not allowed and will be punished. In this way, you could transfer money between to holdings to gain an unfair advantage. So nope 
Direct sale is forbidden, but I think it could still happen, thru a third party. Though I don't think that's the way it should be.
It is impossible to sell shares directly but it is possible and legal through a third party. Currently there are no rules against this when a third part is involved but if you do the entire transfer at exactly the same price which is atleast the market value it should be fine. Just as long as none of the parties involved benefit from it or you transfer money from one holding to an other.
Yes, you can transfer through a third party. One of my partners on Kaitak did that long time ago with multiple companies under one holding when he had two holding on the serve. He either sold the companies or transferred to the holding he keeps running, and no issue.