If all outgoing ports are blocked except 80, is there a way to use port 80 to connect to a remote machine's port other than 80?
I do not have root privileges on the system.
Yes, with help of the third machine that you have full control of and that can connect wherever you want.
The recipe is simple:
at this machine you set up the sshd server that listens on port 80. Then using the putty if you are running windows on your workstation or ssh if you are running linux you are connecting to this machine at port 80. If connection attempts permanently fails provided that you configured everything properly then this means that the transparent proxy is set up at your gateway. In this case you need to set up a simple TCP socket proxy at your workstation which supports connections to upstream proxy server using http protocol and then you get connected with your machine using putty/ssh over this tcp proxy. After you get able to connect to this machine you may tell putty/ssh what tcp tunnels to establish.
Indeed, not really difficult.