Binary Worlds in FT can't be edited in FT.

I don't understand your question about selecting colors in FT. It can be hard to pick an individual color when you have defined many colors, true, but a simple technique is to avoid specifying many colors in the first place. FT interpolates colors, which gives many more effective colors than you selected.

The "Select Coloring Scheme" tab on the Lighting Color window provides a way to save and to load color schemes. You will need to have installed FT in something other than the Program Files directory for things to work correctly because I made a mistake many years ago of keeping user data in the install directory.

Use File>>Export>>Sphere Map Image to write an equirectangular-format image using the current color scheme. I don't think that it's possible to write an image of a precise world size (e.g. Mollweide at 7000 pixels wide) using FT.