Is it possible to have HTTPS connections over proxy servers? If yes, what kind of proxy server allows this?
Duplicated with How to use Socks 5 proxy with Apache HTT
If it's still of interest, here is an answer to a similar question: Convert HTTP Proxy to HTTPS Proxy in Twisted
To answer the second part of the question:
If yes, what kind of proxy server allows this?
Out of the box, most proxy servers will be configured to allow HTTPS connections only to port 443, so https URIs with custom ports wouldn't work. This is generally configurable, depending on the proxy server. Squid and TinyProxy support this, for example.