Miasma will never occur on light tiles, and tiles will remain light even if they're roofed over later after exposing*.
They'll also remain aboveground temperatures, so generally mind that with your water supplies, lest you be forced to also install waving magma heating or have it freeze over.
*Note that stairs prevent exposing things to outside while still allowing creatures, bolts and water to pass. Encasing in ice kills anything that isn't made of fire (or perhaps on fire? not sure) - the military applications are obvious.
As for stacking them on top of each other, you've already discovered water for mass-moving things around. Dump designation - can be applied to individual items, entire categories in stocks screen or areas with d-b-d - will get dwarves to drop things into garbage dump zone (or hole next to it) and forbid them.
But most useful might be minecart quantum stockpiles for you.
Guide.
Though if you don't want to use the corpses, might as well drop them down a hole, inaccessible, and them atomsmash them with bringing raising bridge down on them.
PS: Regarding buying meat off the traders, why not look at your food stocks with z and from stocks menu (note that not everything under "Other" is edible, seeds most significantly) - a dwarf eats 8-9 times a year, so if you have been ordering every kind of meat from the traders and have thousands of it you're likely set for a long time.