That's a great site. Beautiful work. It's definitely an artform in itself. I wouldn't know where to begin to develop dynamic artwork like that. Yes, Grafx 2 and Deluxe Paint both have palette cycle previewing. I like Grafx 2's ability to deal with modern formats, but Deluxe Paint actually has square/rectangular aspect ratio toggles for its tools (except for radial gradient fill, oddly) and a a few more features like gradient flood fills of a dozen or so different styles. Grafx 2 just has circle/rect gradient filled tools with a couple settings and that's it, but it does have capability to work in layers or GIF animation frames. I use both quite a bit for SCI work, though. They're the only good 8-bit image editors with palette tools out there that I've found.
Incidentally, Mark Ferrari, the one who created those cycling images, is the one who worked on some early LucasArts games (made with Deluxe Paint) and is currently working on Ron Gilbert's Thimbleweed old-school styled adventure game that was Kickstarter a little while ago.