Hi, XeDArt! This is really creative, actually.
For animation I would recommend Cosmigo Pro Motion, Graphics Gale or Aseprite, not photoshop. But I'm sure you might make do with photoshop if you really try as well, but if you're going to make pixel assets for a videogame, why not get aquainted with a program specifically designed for this express purpose? Give it a go. All those choices are good, some are more free than others.
As to organizing your color palette, you will find that once you use one of the above programs that problem will take care of it self as they have *indexed palettes* that you can directly alter on the fly. This is a powerful tool for a pixel artist. Want to give your whole scene more contrast? How about just making this and this and that slot in the index darker? Immediately you'll get results in the artwork in a WYSIWYG way. Again, experimentation with the tools of pixel artists and not photoshop will start teaching you certain good pixel artwork methodology tips by itself. The tools tell the tale.
As to your actual artwork, I like it a lot, actually. On your intro screen I would recommend making the outlines of the bones 2 pixels width and perhaps a bit darker, but they can also work as they are with their contour glow if you want. Your clouds are done fast and their shapes could be cleaned up further, but this all depends on how much of a 'clean look' you want your game to have going for it. Same for the blades of grass. You could do another pass and clean a few things up there if you want a more finished and less rough look.
Your dog sprites are quite cool. However it takes a second to parse their faces as dog faces because there's not enough contrast between the middle colour and the shadows that would describe the shapes of the snouts. It's an easy fix, however. As your skill grows as a representational artist, keep in mind this will filter down to your skill as a cartoonist, so it pays to think about construction and light and volumes and shapes LIKE a representational artist, even if your end result will be cartoony. Light that dog face in a simple and straightforward manner, with a bit more contrast, to allow the eye to read it like what it is.
Let me know if you'd like more information about anything I wrote above, and good luck with your game!