Dinobot Sludge’s health is 12. Let’s say there are 15 damage counters total on my other Dinobots. When I flip Dinobot Sludge to alt mode, can I move all those damage counters onto him?

Yes. This will, of course, cause Dinobot Sludge to be KO’d. He’ll be sad.

Does each damage counter moved by Dinobot Sludge's Alt Mode ability create a new instance of damage, or does the total amount you want to move count as one instance?

Neither. Moving damage counters does not count as repairing or taking damage.

If I start my game with Wave 1 Dinobot Sludge and Wave 2 Grimlock and Dinobot Swoop. After I was able to bring the rest of my Wave 2 Dinobots into the KO area, Grimlock is KO’d and Dinobot Sludge was the only card left on my battlefield in bot mode. On my turn, I played the Dinobot Enigma and then flipped Dinobot Sludge to take all of his damage, was this correct? Also, what if I had played a Brainstorm then an I Still Function! and then a Dinobot Enigma after which I would flip Dinobot Sludge? Is that a better scenario?

In your first scenario, Volcanicus appears on the battlefield with 55 damage counters on him. In your second scenario, you can I Still Function! one of the five Dinobots and then use the Dinobot Engima to form Volcanicus. Volcanicus will have 54 damage counters on him (because of the 1 damage ISF! repairs). No matter what, he will be KO’d at the end of the turn, so repairing him with Dinobot Sludge probably won’t do a lot.

Does the I Still Function! + Dreadwing rule apply generally to Combiners or is it specific to Dreadwing’s method of Combination? Suppose I control Dinobot Sludge - Mighty Stomper and have the five Dinobot components in my KO Area. Can I Brainstorm+ I Still Function! + Dinobot Enigma and expect to keep Volcanicus at the end of the turn?

The same ruling applies. If you I Still Function! One of the Dinobots and then play Dinobot Enigma, at the end of your turn, Volcanicus will be KO’d.