@owl, not being there to take a look at your system, I think it will be very hard to help at this stage.
You see, your repositories list file contained some serious problems, which means that at some point you or an application you installed messed with your system in a bad way.
sources.list
is maintained by the operating system and nothing else. And if it got corrupted in the way we have seen (to the point of containing misspellings and missing even the most fundamental of the distro repos) there is no telling what other damage has been done to your installation. It is revealing however that even such a simple thing as installing a browser fails completely on your system.
So, at this point, the best course of action is for you to make a backup of your home directory in case you want to salvage some things and do a complete reinstall of Ubuntu MATE. Trying to fix this problem doesn’t seem to me a viable option at this time. With the loss of confidence on your sources.list file and without knowing when exactly this happened and why, you should also lose any confidence on your system integrity and stability. And that means a reinstall.