As mentioned by Ignacio and Derek, there isn't a single translation for all situations. It depends both on context and the desired tone.
"Confirm" works fine for most of your examples, it's just a little bit more formal than "check". If you were trying to sound sophisticated you'd want to use "confirm" or "validate", but if you were hanging out with a bunch of teenagers, they would probably look at you funny.
To give you an idea on how a different translation can change the connotation of the phrase:
テストを確認する ("confirm the tests")
While any of the possible translations would be appropriate here, "confirm" and "validate" sound more professional, as if scientists in lab coats were performing an experiment. "Check" is general enough to apply to all situations though.
彼と確認して欲しい ("[I want you to] confirm this with him")
ご確認の程よろしくお願いします。 ("Please confirm this.")
Both of these I hear a lot in formal settings, like the office, a doctor's office, or a fancy restaurant. I don't really hear "check" at those places unless I know the other person.
ご予約を確認してください ("Please confirm your appointment")
I think "confirm" is the only translation that would work in this case. When the doctor's office or a hotel is calling to verify that I will show up for an appointment, I've always heard them use "confirm" rather than "check". This is probably because since I'm their customer, they want to be more polite and more formal.
確認方法 ("method of confirmation")
This is the only one that sounds a little funny. As mentioned by Ignacio, "Evaluation Method" would sound better.
