Downtube friction shifters for 11 speed
(a) Friction is gone on the right on 10/11 sp dura ace (they apparently save a spring while doing this, which probably simplifies things a little bit aside from interest in this). Microshift BS-T10 bar ends are 10 speed, front friction with rear switchable, and you can probably mount them on the downtube with a bit of work.

(b) Other friction shifters should be fine. 

(c) SRAM among others do. 

I'd think going with a 9 speed system may be easier (you can find Dura Ace SL-7700 shifters with index+friction for cheap).