On the last hole of my round, my approach shot hits the side of the green and appears to get wet.

I go to the spot I think my ball landed and see a ball just below the surface of the water (no logo or name seen from the position it is lying). Instead of doing the smart thing and take a drop, I decide to do a Jean Van de Velde and hit from the water. Of course, my club hydroplanes and misses the ball completely.

At that point, I decide to take my medicine and take the drop. However, after getting the ball, I realize it's not the ball I hit. I turn around and notice a ball about 6 feet away, still hanging on to the side of the green, near the water's edge. I go up to it and sure enough, it's my ball. I then pickup the first ball that was I swung at and play the correct ball.

Do I take a penalty for swinging at the wrong ball? If I picked-up the ball in the water to be sure it was my ball, would I take a penalty for that too? What would be the proper way to score the hole and if the same situation came up again, could I pickup the ball from the water hazard without incurring a penalty stroke?

stroke play or match play
no penalty for striking (or stroke) at the wrong ball in hazard.

I believe as long as you did stroke the wrong ball that you dropped no penalty

so as long as you finished out with correct ball and did not occur any penalties with it, you occurred no penalties

hahaha
too funny isnt it

so if two balls are in a sand trap and cannot be identified without lifting them, am I allowed to check each ball to determine which one is mine without incurring a penalty shot?

Actually in 2008 the rule changed and you are allowed/required to identify your ball in a hazard, without penalty and you must recreate the lie. If you choose not to and play the ball without identifying it you are subject to the penalty for playing the wrong ball which unfortunately seems to be the case here. You'd pick up a 2 stroke penalty for playing the wrong ball in this situation.

thanks for the update DavePelz4, I was not aware of the updated rules. I learned golf with a stroke penalty for moving your ball in a hazard.

the updated rule makes sense. but why would it be a two stroke penalty and not a one stroke penalty?

Yep! 2 strokes for playing the wrong. You now must identify your ball when in a hazard and you intend to play it. You intended to play it, but it was the wrong ball +2

Two strokes is the standard penalty for playing a wrong ball. At least the stroke you made at it doesn't count