I think that the dye would spread throughout the fibers of the carpet, producing a not-so-crisp edge... but I've never tried it so I can't be sure.
NO! The dye will spread. It's not thick enough to stay where you want it.
I would stick with paint. I don't think that the dye will be thick enough to control and it will bleed, as William said. Also, A lot of outdoor carpets are made of a plastic based polypropylene and they don't really absorb dyes well. Hope that helps.
The other consideration is Rite Dye has to be mixed with very hot water so I don't think it would work. I would use paint with your outdoor carpet.