I will do my best to answer the question, sorry for any inaccurate information. The first sentence I would not say is outdated but should be written as "The best way to form a bond with a pet is to play with it."
"The event will offer you a good opportunity to network with industry leaders from all over the world." Replace network with "connect" and you have a modern phrase.
"Throughout her career, she has battled gender discrimination." This does sound natural but you could replace "battled" with "fought against"
"We're going to videotape the whole event." This does not sound natural and many people from the United States would just say "record" in place of "videotape".
"They winter in their mansion in Florida every year." Replace "winter in" with "go on winter vacation to"
"Parenting a teenager is not an easy thing." No change
"She looked as if she had aged overnight." No change
"We vacation in Hawaii every year." No change
"Yesterday, I lunched with Mark at the Oasis Cafe." replace "lunched" with "had lunch"
"Jacob's twelve sons fathered the twelve tribes of Israel" No change