No. Mirrored domains share all the settings of the domain they mirror, including the SSL certificate. You'll need to set up a separate fully hosted domain to get SSL working.
Maybe. It'll only be cookieless if nothing sets any cookies are valid for that domain… but I'll explain that a bit more in the next part.
For what it's worth, the performance benefits of a cookieless subdomain are pretty minimal unless your site sets a lot of cookies with very large values. And if you're going to go the cookieless route, you should probably consider moving the static content to a CDN as well; the DreamSpeed CDN (part of DreamObjects) is worth looking into.
SSL is fine for those domains, though. No problem there.