I have Ubuntu 12.04 so far, but it has so many problems with wireless keyboards (2.4GHz) such as the "normal" keys do not work and special keys do or "normal" do and special do not. Yes, I tried 12.10 and I think the same. Guess I need to upgrade... So my question is:
Does new Ubuntu (like 18.04, or something like Mint) support keyboards with a touchpad, like Logitech k400 (using 2.4GHz connection)? If Ubuntu 18.04 doesn't have this support, are there any other Debian distributions that do?
Newer versions of Ubuntu have better track/touchpad support. As Ubuntu is available as 'LiveCD', you can try out a newer version of Ubuntu without affecting your current system, and test the functionality of your mouse, keyboard, and other devices before upgrading. To be clear, you won't use the LiveCD to update, just to see what you can expect.
Once you see 18.04 (or whichever version) working, you can use
do-dist-upgrade
to upgrade your system. Using this, and looking up information/issues for each upgrade, you will move from one release (ideally, LTS only) to the next, until you get to the latest version. In your situation, you'll go from 12.04 to 14.04 to 16.04 and finally to 18.04. It's always a good idea to see if Ubuntu has any notes for the specific jumps/upgrades. Normally, there should be no concerns, but sometimes there are additional details or steps.Have you looked into Solaar?
Note: I have a Logitech (Bluetooth) keyboard with a trackpad that has been working since (I believe) 16.04, and works in 18.04. Please try 18.04 using 'LiveCD' and update your question with any specific issues you come across with your specific keyboard.