No need for Firefox/chrome to "download" software, you could get a java-based file manager; this allowed browsing the filesystem and running software on the public library computers, which were far more locked down than any school computer I've ever seen (they had software to enforce 15/60 minute session limits, session reservations, etc, and group policies that disabled external storage, explorer, and the start menu; most browser functions were disabled).
If you've got a Citrix-based application available, there's usually some way to trick your way into
that system; in my case it was by going to file->open in the weird processor, and right-clicking it to open it in (the remote session) explorer, which had my local media mounted (flash drive), and allowed access to the remote browser. If you can get a cmd prompt, you can also sometimes get around restrictions with that, possibly by using it to open task manager. In college, these weren't locked down so much as intended to be single-purpose (for running Photoshop, etc from home), and I used similar tricks to gain access to a command prompt so I could register fonts for my projects using FontLoader[1].
Of course, despite every PC being on a domain, there was still the local administrator account, which was easily obtained by leaving an unused computer running ophcrack overnight. Eventually I found out that the roaming network administrator account password was simply ford, as the admin left a local account on a computer that wasn't networked; this pretty much lasted me through high school. For bypassing the firewalls? That was simple; I got a free shell account, and later my own server, an early openvz vps.
Once I got my own server, I spent long amounts of time trying to increase the amount of RAM available to me (since it was fair-share CPU on a decent server, building and similar jobs ran far faster than my current digital ocean VPS). At this point, I found out about sshfs on my own computer, and spent ages trying to find a decently cheap Xen VPS so I could use kernel modules for sshfs and swap. Eventually, I gave up. One period of my last semester I was a TA, rarely with any work to do, so I spent most of my time on IRC and setting the wallpaper to be Cyanide & Happiness comics.
/me trouts Piper around a bit with a large slap
1: this used to be available at https://bitbucket.org/cryptw/fontloader but not anymore. This might be the source, not sure. https://github.com/stevenstrike/FontLoader