When I install Firefox, it wants me to replace the older version with the one about to be installed.
Seems like an obvious question, but can I have, install, and run, two different versions of Firefox on my computer at the same time?
This should be possible. When you download FF (and I assume that you are doing that from Mozilla directly, rather than having FF update itself), move it to the desktop. Then rename it, perhaps Firefox Latest (or whatever you like). Then, move it to /Applications. It should happily coexist with the previous version in /Applications.
Jon is mostly correct.
I usually keep 2-3 versions of FF in my Applications folder (labeling them by version number). And I keep all the .dmg files (at least back to FF 8).
They can stand side by side but only one will (or should be) run at a time.
Older versions will use the most current 'helper' files, including your Add-ons (plugins and extensions).