1) I hit a ball to the base of a tree that had a mound of wood chip mulch protecting the base. My ball was resting against the wood chip mulch in a way that I would not be able to follow through on my stroke. The tree itself was not an issue. Can I move this ball back a club length without a stroke penalty so I can take a normal swing? This scenario ended up costing me three strokes on a par 4.

2) After heavy rains, I sliced my tee shot over a serious puddle that isn't a declared water hazard. The ball rested on a good piece of rough about 2" to the right of this deep puddle. I could not address the ball without standing in water that was deeper than my ankles. Can I move this ball a club length to the right (away from the hole) so I can address the ball and stay dry?

3) Tee shot hooks and comes to a rest against the junior tee markers about 200 yards down the fairway on a par 5. Can I move the ball a club length back or to the left (away from the hole) without incurring a penalty shot?

4) My favorite. After hitting the edge of the green on an unfamiliar course, I find that there is a sprinkler head on the green that my ball happens to be sitting on. This sprinkler head is right at the edge of the green, and moving the ball away from the hole will effectively move my ball off of the green. What do I do?

5) Finally...more heavy rains have turned a sand trap into a water hazard. I do not know this when I tee off, and I happen to plop my ball right into the middle of the water. How can I play this?

Wood chips are part of the course. They are considered loose impediments. No relief.
Puddles are CASUAL WATER and you ARE entitled to free relief. In the hazard, you may move the ball without leaving the trap. Do not improve the lie. You must move back.
If you can move the tee marker, by all means do so.
Never came across a sprinkler ON THE GREEN.

Ya...the sprinkler head on the green was a little unexpected, but the course itself was pretty ghetto. I ended up moving the ball back, off the green, and chipped over the sprinkler head. There are a couple of ways to interpret that. Had the sprinkler head (and the resulting indentation) not been there, my ball would have probably rolled off the green anyway. Had I played my ball from that spot with no sprinkler head, I probably would have had one stroke less. It's a toss up.

I never even thought about moving the tee marker. Duh.

The wood chips sucked. I had to take a safety shot to the fairway, which I totally biffed into another tree. I then biffed that shot into yet another tree before I had a clear shot at the green. Could I have moved my ball a club length back with a one-stroke penalty?

I played the other two scenarios correctly.

You can declare an unplayable lie from any location on the course. Two club lengths no closer to hole plus a stroke.

For the sprinkler head situation, you get free relief no more than one club length from the point of closest relief, no closer to the hole. So you don't always have to go back, sideways is ok as long as you're no closer to the hole. The important step is to find the point of closest relief.

For the water in bunker situation, if you can't find a dry spot within the bunker, you can drop outside the bunker (with a one stroke penalty).