Interesting-- I've got a bottom bracket kit I've been working with. The only thing I haven't replaced at this point are the cups (I couldn't seem to get them to move ever with hammer and screw driver, though I'll admit I didn't give them a really hard whack either). I'd been trying to mix and match crank set parts in order to get the original look (double dust cap) but with new innards. What I kept noticing was sort of a cyclic "hiccup"-- when the right arm would get near the bottom of its travel I'd feel just a little "bump" with my feet. I figured the source was mixing and matching the parts, so now I've gone over completely to the new set, except for the cups.
When you say there should be a washer between the sprocket and the races, you mean I take the washer that came with the kit and: slide the sprocket onto the crank, then slide the washer over that and then tighten the cone down on top of that, then put the bearings on top of that and slide it into the bottom bracket? When you talk about "clearance" issues, do you mean that the sprocket will catch on the frame or bottom bracket? I installed it without the washer in this last attempt, though using the washer left the chain line looking a bit weird.
The equipment I'm using is this "tange" bottom bracket kit that I got from a bicycle shop. It seems to be quite generic, though the parts seem generally to match what the original stuff was.