The last finding makes me suspect the clutch or the 2nd gear cogs. If it goes away when you disengage the clutch and you're still rolling that lets the rolling gear, chain thru wheel, off the hook. Doesn't necessarily eliminate the tranny because it isn't loaded, just spinning. Might just have a worn bushing or shim that needs replacing, or, the clutch springs could be out of adjustment causing a tiny wobble...that would be my first look-see.
