I'm pretty sure that bit that's 'stuck' in there is the part of the shifter the cable head sits in - left and right aren't made the same. You'd be lucky if these shifters still work properly, but a solvent flush and re-lube with heavy oil might make them good again. I think maybe the 9s ones have other typical failure modes beyond the old gummy grease though.
And yeah, the fork won't stay in the frame without either the stem clamped or at least the top cap fitted, unless the wedge ring is stuck.