Question 1: Do desktop environments come with a default set of programs?
If I uninstall Mate and install LXQt will I end up with different programs?
Question 2: How do I go about installing software that is not in the optimised software list? (Geany and Kid3)?
basically there is not that much installed on any desktop, except links to some DietPi scripts and Firefox as browser.
Software that is not part of DietPi software catalogue, would need to be installed following software vendor installation instructions.
Since DietPi is set up on a regular Debian/Raspbian OS repository, you can follow regular instructions for those and most are available through the regular APT repositories:
apt install geany
apt install kid3
For RPi/Raspbian it is the same in 99,5% of cases as it is basically a Debian-clone for armv6hf.