Keeping MuseScore 2

You can use versions 2.x and 3.x on the same computer at the same time. The internal format of the files changed significantly from version 2 to version 3 so version 2 cannot open version 3 scores, but version 3 was programmed to make a reasonable effort to open version 2 scores so they look OK as long as you haven't done too much customization of the score. I've had a lot of success transferring my scores from version 2 to 3 by opening them, pressing ctrl+a then ctrl+r to reset everything to its defaults. Though some staves are different widths than they were previously, you can usually reset the system breaks to your liking without too much trouble.

You may as the IT department if they actually deleted version 2 or just hid/deleted the shortcut to it and ask them if they can restore your ability to use version 2.

You can indeed keep both versions on the same computer, and this is recommended if you have a lot of older scores with extensive manual adjustments to achieve a particular layout. In general, default layout is greatly improved in MuseScore 3, but as you are seeing, older scores highly customized (often to work around quirks and limitations in MuseScore 3) won't look the same. it's usually possible without much effort to update them to look even better in MuseScore 3, but it does take a little while to get comfortable with that.

Not sure what you mean about the "architecture" of MuseScore 2 vs 3. There were some very significant improvements in the default layout, as mentioned, and some other improvements in other areas, but it's still largely the same code. The file format has changed to the point where a file saved in MuseScore 3 cannot be opened in MuseScore, so do keep that in mind.

And in case your IT department refuses to install MuseScore 2 again, you could use a PortableApp version of if, running from a USB thumb drive