Can I run a SRAM road rear derailleur with downtube shifters? Can I run a SRAM road (apex/rival/force/red) rear derailleur with downtube shifters?  

I though you might be able to mount the bar-end shifters on downtube, but it seems like they are a ratcheting type as opposed to the usual downtube shifter design.

I've also heard that shimano shifters will probably not pull enough cable in friction mode to accommodate the entire cassette.

Is there any way to do this properly?
If you set the downtube shifter to friction mode then you can run any rear derailleur you want. If you're looking for indexed downtube shifting with a SRAM rear derailleur, the answer as of the date of this posting is no. Cable pull for SRAM is different than it is for Shimano and others, which is why mixing different brands of shifters and derailleurs is not advisable.