both of my placentas were delivered within 15 minutes of my babies coming out. we cut the cords after the placentas were out. as far as getting it out; it's easier. i mean, yeah your tired, and your yoni hurts, but there are no bones in a placenta :P
sometimes it's messy, but usually a birth attendant will catch the placenta, and all that follows it (blood), in a bowl or on a chux or something. usually when i'm at a birth the messy part is what we call the "separation gush", which indicates that the placenta has separated from the wall of the uterus and will come out shortly. usually at that point, mom is still in the tub or we've moved her to a couch/bed with a chux under her to prevent blood from getting on something it shouldn't.
as far as planting it after; i think planting it with a tree would allow for you to plant the whole thing as opposed to cutting it into pieces. that being said, my midwife's partner gardens and will plant whole placentas in garden beds (some clients don't want their placentas) without killing or causing the plants harm. in fact their garden boasts some mighty beautiful flowers and herbs!