Ubuntu comes with the Unity graphical environment.
Canonical makes Ubuntu and Kubuntu.
http://en.wikipedia.org/wiki/Kubuntu
http://en.wikipedia.org/wiki/Ubuntu_(operating_system)
http://en.wikipedia.org/wiki/Canonical_Ltd.
Linux Mint is a fork of Ubuntu. Well, it was, now you can get a pure Debian based version if you like.
http://en.wikipedia.org/wiki/Linux_mint
Canonical has been criticized lately for actions that may not be in the spirit of openness and open source.
If you're worried about keeping old hardware running quickly, lubuntu might be the choice for you. Lubuntu is a lightweight Ubuntu, I've been very pleased every time I use it!
http://en.wikipedia.org/wiki/Lubuntu
None of these operating systems are inherently better, however one my suit your needs more than the others. Since the linux flavors listed are all really Ubuntu at their core, you can expect them all to run the same software without much issue.
Your computer absolutely can run any modern version of linux.
sorry, gotta goto work. I'll address the rest of your questions once im in my office.
EDIT: And here we are!
These versions of linux, and most others will run Libre Office without issue. Libre Office works great! It will open Microsoft Office documents and has everything you'd expect out of an office suite.
Yes, you can partition your HD and set up a dual boot system. This will allow you to choose which operating system to use at boot time. You can set this up in the linux installer.
You will need to research the copier in question. Many manufacturers make complete drivers for linux, some do not.
Being linux, there will be snags. Research and more research before you try something will help tremendously. The community also does a pretty good job of helping out. If you're comfortable using the command line periodically, then you will be fine.
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