I just cured what you've described about your headset. Mine is a Russian size so I can't get a replacement headset. Headsets wear more at the front and back than at the sides. Take the fork off and pop the two headset cups out the head tube. Put the top one back in 90 degrees round from where it was and the bottom one 90 degrees the other way. You've now moved the worn bits out of phase with each other and it'll now turn freely and smoothly when re-assembled.
Run friction shifters with little or no lube, otherwise they do drift unless they're done up so tight they won't operate. Check the limit screws on the front mech as well, because if it's marginally too close it'll keep jumping off the big ring if there's no indexing to hold it in place.
Sounds like your seat post might be too small, or there is too much grease on it.