If you just need vegan food coloring, most brands out there don't contain any animal ingredients. The only one you have to watch out for is "carmine" which is bright red, but is also found in pinks, purples, and even blues (I've seen this mostly in powdered colors). There's a brand of natural food colorings that Fresh Fields/Whole Foods Martket uses in their bakeries. Even when I worked there, I couldn't find out if these contained carmine or not. There are no labels on the bottles and no indication of a company name. They were just ordered from a large vendor. Williams-Sonoma carries some natural food colorings (which I'm pretty sure the red has carmine).
Anyhow, back in the day at Fresh Fields, before we found the natural colors, we made our own dye with some of the very things you mentioned.
Tumeric = yellow
Paprika = peach
Beet juice = pink
Liquid mint flavored spirulina = green
Blueberries = pale blue
Blackberries = lavender
Cocoa powder = brown
You can also use raspberries for pink. For the berries, we boiled them in the microwave and strained the juice through a coffee strainer. You should freeze the unused juice as it goes bad in the fridge rather quickly.
Except the brown, you can't get vibrant colors at all. They come out very subtle shades of pastels. The tumeric and paprika don't altar the flavor b/c such minute amounts were used. I think paprika is better than annato. I just bought a pack of annato seeds and ground them into a powder (which not all of them would turn to dust- had lots of clumps). Let me just say that that stuff SMELLS! It's horrible! I didn't even want to bother putting it in icing. Too darn stinky!
Have you seen any good prices for beet juice powder. One web site was selling it for $22 for 1/2 pound. I'm sure there must be something cheaper out there.