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
No, you need wool.
No, they do not.
You can dye wool with it, or you can create decorative blocks of it using 9 pieces.
string
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.
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.
You can't. You get string from killing spiders. However, you can craft string into wool.