Yes, you can use the Windows GitHub client and the GitHub Desktop client with GitLab, BitBucket or any other hosted Git solution.
We only use it with HTTPS and you'll need a valid certificate if you do use HTTPS. It may work with HTTP as well. We never did get SSH to work completely right since it's a tough to inject your own SSH keys into the application.
If you want to clone a repository, you have to drag and drop the HTTP URL onto the GitHub application.
I was unable to get the drag and drop trick to work on OS X. But you can add locally cloned repositories into the OSX version and then the application works like normal. And OSX supports SSH keys unlike the Windows version.
The GitHub for Windows application allows you to manually add and work with repositories from any remote location (ie GitLab/BitBucket...) or local location (your hard drive or network folder).
However, it only has the option to add your account info for GitHub. Whatever functionality that is tied to that account login will only work for GitHub. I am guessing that functionality is automatically recognizing all your GitHub repositories.
Yes you can use GitHub For Windows with GitLab, You can even use SSH. (The accepted answer didn't get SSH working, but here is how you get it working.)
Add your public ssh key from github_rsa.pub (which is found in your .ssh folder) to your SSH keys on Gitlab.com
Add a config file to your .ssh folder that looks like this
Host gitlab.com
RSAAuthentication yes
IdentityFile ~/.ssh/github_rsa
User mygitlabloginemail
Clone / pull your repository through Git Bash
Drag the folder to Github For Windows
And thats it, you can now use Github for Windows with your gitlab repository.