I did the Ancestry DNA (I did it for genealogy reasons). Anyway I discovered I had a half sister she was adopted in the 60s in America. Never knew anything about her. My father died in his 40s and he never knew about her either. It was a huge shock but an amazing find. I’ve since met her.
In regards to should you get your dad to take the test. I would wait and see what shows for you first and if you have a close relative appear then get your dads tested and you’ll know how close the relationship is. Best of Luck with it.