I would use burst for volleyball. I assume you're from the side and not too close, in which case the people aren't really moving around all that fast, and the ball is going way too quick for your AF. I'd just use burst and pick my favorite shots from the bunch.
I don't know what dog trotting is, like a dog show? I'm sure any method would be fine.
I don't use in camera noise reduction because I prefer to do it myself on my computer. But if you don't do post processing then give it a shot. So long as it's not too strong you should be able to use it all the time. I'd guess that the camera uses far less when there isn't much noise (i.e. shooting at low ISO). Take a couple shots with and without, in a darker situation, and see how you like it. If the camera applies too much NR there will be a lack of detail, especially on people's skin. It'll look plasticy.
I'd always use the highest resolution my camera offers, which is probably 'super fine'.
