Should somebody with plantar fasciitis wear high heels?

Yes and no. The heels will take a strain off your achilles tendon, but will not be very supportive in the arch or rearfoot. A good supportive structured shoe is best when treating plantar fasciitis: think of it like a brace for the arch of your foot. Like with any injury, you want to brace the area as it heals, whether that's with a sling, cast, or in this case, a supportive athletic shoe with orthotic.
No. You want to stretch the back of the leg in the other direction. Toes up, heel down. See your podiatrist !