I'm not much for introductions, so I'm just going to jump in right here…
I've converted my old Varsity to singlespeed/fixed: new flip-flop hub, freewheel, fixed cog, chain, and chainring. I kept the original one-piece crank rather than using an adapter and installing a modern three-piece.
Here's my current puzzle to solve: The drive pin on the right crank arm is slightly smaller than the drive pin hole on the chainring. This causes the crank to slip slightly forward or backward when I resist the direction of the crank arms while riding on the fixed cog. For example, if I'm riding along forward and then resist the forward motion of the pedals, the crank slips backward until the drive pin meets the back side of the drive pin hole in the chainring. After I have stopped in this manner and I begin pedaling forward again, the crank slips forward until the pin is against the front side of the hole. (Of course this is never a problem when riding the freewheel because there's no resistance when backpedaling.)
The way to fix this seems to be to find some kind of bushing or washer that will fit around the drive pin to shim it up inside the hole in the chainring. The gap is very small, less than 2 mm, but I feel it in my legs every time the crank slips. Maybe someone here has done a similar conversion and has found a solution.