There are nonstops from SFO on ANA, Northwest, and United. There are also flights from LAX on Korean and JAL. JAL is part of Oneworld, however, so I'm not sure you can do a transfer that would let you get a JAL flight.
The most straightforward way is to transfer to ANA and book a flight on them.

For Northwest or Korean you can transfer miles to Delta or Continental and book a partner flight. The downside is that partner availability is usually less than on a carrier's native program.

For United, you can transfer miles to ANA or Air Canada. The same caveats apply.

You cannot return miles to MR once they're transferred. Depending on the timing and on whether they allow a hold, it's possible the seats could no longer be available by the time you transfer the miles. Miles to Delta transfer more or less instantly. I don't know about the timing on other airlines, but it can take up to a week.

To get a sense of what the upper classes are like, you can poke around the airline-specific forums here. Very generally, service is better on non-US carriers, but I've never flown ANA or Korean myself.