I actually take objection to the idea that either of these would ever have happened.
Kirk died at the bottom of a tall cliff - twice as tall as the one they used for filming, thanks to clever editing. There is no way the frail Picard could have hauled that carcass up to the top of that rock formation, along those sheer walls that had but near-vertical metal ladders for access. Not even a young fireman in his prime, rested and properly hydrated, could have done it. Picard didn't as much as have a canteen to bring him up to speed.
Moreover, the pile of rocks Picard created was too small to hide the corpse. Unless, of course, Picard hacked it into tiny pieces and carried those up one by one. And then washed his uniform with the water that wasn't there.
Hell, getting Kirk separated from his bridge would have required inhuman physical effort (and that missing water) already. No, more probably the corpse was still down there when the shuttle arrived, and all that Picard took to the top of the cliff was Kirk's badge. Going to the top would be a logical maneuver (getting the lay of the land, trying to attract the attention of rescuers Picard knew would eventually be there) - and taking Kirk's badge might not have been a mere empty gesture, either, considering that such badges were communicators in "Yesterday's Enterprise"! After Picard found out that Kirk's badge was but a trinket, the older captain would have had nothing better to do than erect a little discreet memorial...
As for Spock, the torpedo wasn't supposed to soft-land - this was a surprise to the Grissom team, attributed to the local gravity being in a state of flux. We could just as well assume it wasn't supposed to land at all, but merely curve around the planet in the most spectacular manner possible.
Timo Saloniemi