The difference between fresh cut live logs and standing dead wood is one of inoculation. You can be pretty sure the former is sterile and uninfected, and you have no idea what kind of spores have blown in and taken up residence on the latter. If you drill holes and pound your dowels into something that is already infected, you may be setting up the battle of the hyphae between different competing fungi. Who knows if the type you want to grow will win that competition.
Definitely don't bury them, that just brings soil fungi into the picture and makes it a multi-way competition.
Shade is good, completely out of the sun is even better. If it sees any daylight, then algal spores may start to colonize the log as well, and instead of a pure fungal culture, your log could have a mixed algae/fungi combination growing -- a lichen.
How ever you store them while the fungi are growing, remember that fungi need air and water -- enough air flow to get needed oxygen, but not so much that it dries out.
I'll let someone else take a crack at which wood goes best with which mushroom.
I agree with everything that John said. Using freshly cut, healthy logs is crucial. And keeping your logs off the ground is important, especially for shiitake. (After the logs are completely colonised, the oyster logs can be sunk into raised beds using the totem pole method, but not the shiitake.) Cut your logs in early spring before the buds open, wait two weeks, then inoculate.
Silver maple, cherry and especially oaks are good for shiitake. Hackberry, box elder, cottonwood and elm are good for both kinds of oysters. (Although elm is actually best for the golden oyster.) Pines are usually not good for mushroom cultivation. I've had excellent results with shiitake on oak and oysters on cottonwood. Some of this info is from the Field and Forest catalog. Good luck!
"In a fruit forest everyone is happy"- Sepp Holzer
In addition to keeping them in the shade be mindful of airflow too. Dryness is the enemy. Somewhere not totally stagnant but with little risk of drying breezes is perfect. If you have brush growing around you could always cut a little 'tunnel' hole into the brush and stack them off the ground on a pallet.