What sustains Watermelon Stevens?

Magic, of course!

OK, let me elaborate. We have seen more evidence of magical minions created by a Gem, some of them out of thin air and others by controlling existing creatures. Some examples are some of the corrupted gem's “offspring” (such as with the Centipeetle and the Crab), Rose's moss and bushes, and Holo-Pearls. Regarding both the corrupted gems' offspring and Holo-Pearls, they actually survive until they are poofed; and Rose's plants are seen active long after Rose's “death”.

So, with this evidence, it's safe to assume Watermelon Stevens work in the same way, remaining active until somebody kills them.

Most magical things stop after the Gems stop concentrating on them, so why are the Watermelon Stevens able to keep moving and even create new versions of themselves?

Not quite. The magic weapons do disappear after some time, but all magic beings we've seen so far continue existing on their own.

Why are the Watermelon Stevens so kind and decent (except when they went to war against Malachite)?

That would be because of Steven.

Remember that after the Watermelon Stevens attacked the Crystal Gems, Baby Melon sacrificed himself to end the disaster, and Steven tells them to go away until they understand what Baby Melon did.

Steven: Look at what you've done! You think I wanted this?! You should all be ashamed! He knew this is what it would take to calm your rage! ... He understood true loyalty. Steven: Go! Think about what you've done! And don't come back until you understand what he did for me, for all of us! (Steven Universe - S01E34 Watermelon Steven)

Since the Watermelon Stevens go away, we see that they did understand him, and as a result, the next time we see them, they're only aggressive towards Malachite, the real threat.