I've noticed that many graves don't have any flowers planted within the allowed proximity of the tombstone (each graveyard has different rules). I hardly think the dead would mind if we gave them some respect... and who's to say that that old neglected grave isn't one's long-lost relative? How about some poppies for the war vets or some perrenial forget-me-nots? Whatever is chosen, it has to be short and not likely to become invasive, so roses, even miniature ones would not be a good idea. Since it is traditional that headstones face the rising sun in many cultures, shading is only a problem when there is a tree or shrub nearby.
To respect closer relations, I think it would be considerate to wait to plant until after seedlings or transplants planted by closer relatives are showing.
A cemetary gardener near me told me that adding bonemeal to the earth will deter squirrels digging up the plants.