Fixed mountain climbing gear. Help please

+1 about the flip-flop, considering doing the same up L'alpe D'Huez this summer or next.

ETA - "From here, the ascent to Mount Dalsnibba starts. This last bit is 5 km on gravel road, and has a climb of 1:8."

Is that do-able fixed? I'd imagine traction would be a problem, one bit of wheelspin and all your momenteum is gone.

Have the same concerns regarding gravel roads :(
I am considering a rear tyre witha bit more tread pattern.

If I'm running say 42:22.
Is it worth getting off the bike to flip to say 42:23.
Is there such a thing as a 23T cog?