Dyes wouldn't be any good for creating images,and the sort of fabric paints you would find in a cheap store are designed for little children to do t shirts and they fade away really fast, If you want do do some proper creative work you need to select a type of fabric paint that is appropriate for the fabric you are using, Dylon fabric paint is fine for cotton or poly cotton but you need special paint for silk and for silk you also need an outliner to stop the colours running together, I find it easiest to buy them online from Fred Aldous.
Yes you can mix them so you only need to buy a few colours Red, yellow, medium blue, black ,white , and if you want a proper purple you will need that too as the red/blue mix tends to be a bit plummy
If you want to paint on black however you will struggle as you will need to mix enough white in the colours to make them show, the gel type fabric paints in tubes are better on black as they sit on the surface, TEXTIL markers are brilliant for fine work as they come with a thick and thin end
I use Setacolor and setasilk
and no you cant use your hair straighteners
A) because it is downward pressure that makes the dye set into the fabric and
B) Because even if it doesn't say on the jar/tube the best way to finish off is to iron the finished design through siliconised baking parchment Do not use greaseproof paper it doesnt work and just peels the image off the paper.
Hope that helps, I will check back on this thread in case you need more help