Thanks for checking.
Short answer: Use a maximum of 16 colors, chosen from the 32 available.
Long answers:
1. To keep things simple, you can use up to 16 colors in your game. The color palette will have 16 colors that you can choose from, but you could limit yourself even more if it fits the style of your game. For example, if you want a 1-bit style, you may choose to only use black and white, or if you want a Gameboy style, you may choose to use only 4 greens. You can get some ideas for limited palettes on Lospec and then try to find the closest matching PICO-8 colors for the visual style you like.
2. Yes, you can use any of the 32 colors that PICO-8 has to offer (16 Default + 16 Hidden). However, you should choose only 16 out of those 32 colors and stick to drawing all game sprites to only those custom 16 colors. It is possible to swap the palette around using code so that your game could even use all 32 colors, but those require advanced techniques and have extra limitations that I wouldn't recommend worrying about here. Choosing and sticking to your own custom 16 colors is quite simple and straight forward though. PICO-8 Palette Maker is a nice tool for selecting a custom 16 out of all 32 colors.
3. Sticking to the default 16 colors is the easiest path, and I suggest that for anyone who doesn't have a clear idea of what your game will look like. But if you do care about the visuals more and want to create mockups true to your imagination of the final design, then I suggest making a custom palette. The limitations of a custom palette are as I wrote in #2, choose any 16 colors out of the 32 available, and stick to those 16 or less in all of your art.
Custom Palette Example:
Here is an example where I used Palette Maker to choose 16 colors, copied the code to create that custom palette, brought it to PICO-8 Education Edition, and applied the palette to even change the editor colors so I can draw the sprites and map in the custom palette. See Example