Running Docker under Windows/cygwin environment
On my cygwin environment the docker cert path is actually set as below and docker seems to work fine.
DOCKER_CERT_PATH=C:\Users\user\.docker\machine\machines\default
The following does indeed give an error
DOCKER_CERT_PATH=/cygdrive/c/Users/user/.docker/machine/machines/default
$ docker version
Could not read CA certificate "\\cygdrive\\c\\Users\\user\\.docker\\machine\\machines\\default\\ca.pem": open \cygdrive\c\Users\user\.docker\machine\machines\default\ca.pem: The system cannot find the path specified.
So try changing your DOCKER_CERT_PATH to regular Windows path format.
export DOCKER_CERT_PATH=C:\\Users\\Alexey\\.docker\\machine\\machines\\default
My docker version is 1.10.1, if the results are any different.