Two things.
1. You can execute cygcheck by navigating to C:\cygwin or C:\cygwin64
in Explorer and opening a command or powershell window from explorer's
shift-context menu. There, run the following command:
.\cygcheck.exe -csr > ..\home\CYGWINUSERNAME\cygcheck.out
or similar.If this does not work, I would look for a anti-virus or
firewall that is interfering.
2. I don't recall you mentioning how you start cygwin. Please try both
standard methods: a) execute c:\cygwin64\Cygwin.bat and b) executing
the Cygwin icon that (by default) cygwin setup places on the desktop.
3. As an after thought, I always run cygwin with CYGWIN_NOWINPATH=1
set in my system environment variables. (From what I understand, I am
one of the few people that do this!) This resets the PATH variable in
the cygwin shell to only contain cygwin directories. If this resolves
the problem, you can edit your .bashrc to add any needed windows path.
For instance, I add the path to some go lang executables. And I set up
aliases or functions to run a few specific Windows exes, such as
Explorer, icacls, cmd, etc. If you want all the directories from your
windows path, they are available in the ORIGINAL_PATH variable, so you
can add all of them at the end of the PATH variable, so they do not
shadow same-named cygwin exes.