First, you have to find sheep with wool. Then, use your dyes to dye its wool to a certain color. Then, kill it or shave it to get the wool.
No, you need wool.
No, they do not.
wool in minecraft
string
You can use wool for making beds and paintings. I'm not sure about the painting but I know you can make a bed in Minecraft PE
you can make beds and colored wool if you find other things
You have to put a cactus in a furnace and then apply the resulting dye to the wool.
Kill a sheep or get some sheers
For colouring wool blocks.
there is no "carpet" item/block in minecraft, but people use different colored wool as carpet for minecraft.
You can't. You get string from killing spiders. However, you can craft string into wool.
You can dye wool or sheep black with it.