Ssh port forwarding freezes after 10 simultaneous connections

There are two parts to getting this to work, so I'll address them seperately.

Connecting to your Server:

As you've mentioned that all you can talk to is a proxy on 443 and 80, you'll need to tunnel your SSH connection out through the proxy. You do that by telling Putty to use the proxy server to connect. Under the 'proxy' menu. Select HTTP and then enter the details for the corporate proxy.

From your post, it seems like you've got the connection working fine.

Configure Putty to Create a SOCKS proxy on the local machine

Both putty and OpenSSH support opening a SOCKS proxy. For OpenSSH you'd use:

ssh -D <port>

And then point your browser at that port. In putty you create a 'dynamic' port forward. You'll find it under the Tunnels menu. Enter your desired listening port and then enter anything you like as a destination (it gets ignored for dynamic forwards). You can then point your browsers proxy settings at that localhost:<port> and it should work.

For more information, the relevant part of the putty documentation is at http://tartarus.org/~simon/putty-snapshots/htmldoc/Chapter3.html#using-port-forwarding.

As others have posted you may not want to circumvent the corporate filtering though as it's not something you may want to explain :).

You can do the trick using some Perl:

sslh - Switch incoming connection between SSH and SSL/HTTPS servers