1994 Honda Nighthawk Idle too High & Clutch Question

I have a '94 Nighthawk 250 that I bought from a friend who let it sit
for a few years relatively untouched. When I got it it barely ran.
One of many things I did to bring it back was rebuild the carb. The
bike ran great for about a month but in the past few days I've noticed
the idle is kind of high when stopping at a light (though not nearly
as bad as before rebuilding the carb). I thought it could have
something to do with the choke being stuck or something so I played
with the choke knob and I found when I pulled it out the idle dropped
at one particular spot - if I pull the choke further or push it in all
the way the idle seems to go back up. Any ideas of what to look for
on this?

The other thing is sometimes when I downshift (especially when coming
to a stop) pushing the shifter pedal does nothing. I let out the
clutch then pull it again and then it'll downshift. Is there
something not quite adjusted right? Is there something I can adjust
for this?

The bike needs to be rolling for downshifting to work. If you're still in
a high gear and almost stopped, you're likely to have this problem. If
you're stopped, letting the clutch out to the friction point and pulling it
back in will let you shift down. This is is normal. I'd suggest
downshifting while the bike is still moving at a substantial speed and not
waiting until the bike is almost stopped.

The bike has somewhere between 5000-6000 miles (the odometer is
broken, but my friend told me it had roughly 5K when I got it and I
know I haven't put 1K more on it just yet). That's something else I
intend to fix, I just haven't got to it yet. Actually, anyone know of
a source for a used one that works or are these repairable?

I really doubt the gearbox is broken, given the low miles and the fact
that this engine as rock solid.

First, have you tried adjusting the idle speed rather than pl,aying with
the choke?

Second, is the gear lever securely bolted to the splined shaft?

Thanks for any help or ideas!