1) The matter is complicated. Father has no objection when it comes to permanent marriages, but he take mutah as a bad thing. He has no objection if the girl talk with a boy or even goes out with him. He has no objection in all these things.
2) Is bikni waxing(including pubic area) is valid?

1) You can speak to the boy and get to know him and getting to know another does not require temporary marriage or holding hands. If the conversation with a boy is limited only to get to know him it is not haraam. Refrain from provocative and sexual conversations and there should be no problem.

2) Looking at the private parts of another is not allowed. Usually when bikini waxing is done, the private parts are seen and this is a problem.