Not recommended. The paint will lose durability and sheen as well as most likely become chalky.
What you can do is thin down the joint compound a bit with water roll that onto your wall to texture. Allow to dry fully, then prime and paint. The difficulty in applying texture with a roller is getting it uniform and even. Unless you're trying to match an existing texture, the use of a texture gun is a better way to go. Texture guns are cheap to rent or buy.
Yes......but.......whether you should is another question.
IOW, it depends.
What's the game plan?
Faux finish of some manner?
If you are attempting to create a venetian plaster look using tinted joint compound, please take my advice and DONT!! It is quite possible to tint your joint compound, it will make whatever color you add to it much lighter.. however when it dries, colored dust will wipe off the walls for eternity. It will not give the dimension, or sheen that the marble dust in ven. plaster gives, and the look is about like using matte finish paint and colorwashing a wall. Also when you sand the plaster, everything will be coated with colored dust which is even more of a pain to clean.