1. Are there websites with up-to-date schedules for numbers stations
and/or pirate stations?

i think restricting to your search to clandestine communications might work.
there are whole books written on "secret" things.

2. Are there any stations that can be heard from the Binghamton NY
area?

certainly. i've heard numbers stations in Tennessee and Japan.

I know both numbers and pirate stations are a little secretive, so it
seems unlikely that either of these would have known schedules.

they operate in certain bands. theoretically you could record a big chunk
of each band, run it through spectral analysis software, and find stations.
the numbers stations are doing stuff like "1 2 3 4", which is highly
repetitive.

I found quite a few web pages with broken links to pirate radio station
pages, and I found a few that were still (apparently) operative, but
they were broadcasting from California on frequencies that seem likely
to be overpowered by closer stations.

superstation yacht pirates are mostly gone? there should be some landbased
stations.

Spookish Gravity