Creating a Git repo at / (root) for tracking settings?

The answer to both of your questions is no, you can create any directory you want in the /. the only thing that could happen is some permission issues with some spacial paths I guess.

However it's better to store the .git directory somewhere else, something look like:

git --git-dir=/home/user/backup-root --work-tree=/

Read here.

Actually, you probably want to version control configuration files in /etc/ (you don't care about the entries of the root directory /, notably directories like proc or usr or bin in /) so you may want to install the etckeeper package

And you could also version control some selected subdirectories (like /usr/share/applications/ that you mentioned).

However, don't mess with the Ubuntu package management system. Perhaps you mostly should backup the current list of installed packages.

Having a git repo in / works fine, except that it does make it hard to notice when you have a lower level git repo with problems as it will answer for all of them.

Note: It is less work and probably more useful to use 'debsums'

sudo apt-get install debsums

Which will allow you to quickly detect (most) changes in binaries or in configuration files.

As an example from the installed packages here are the ones that differ from the upstream packages.

$ sudo debsums -c
/boot/vmlinuz-4.10.0-19-generic

And you can get a list of changed configuration files with:

$ sudo debsums -ec
debsums: missing file /etc/default/chromium-browser (from chromium-browser package)
 /etc/libvirt/libvirt.conf
 <snip>

Note how the chromium-browser is improperly packaged and and has a file listed in the the package file list that doesn't exist.

/var/lib/dpkg/info/chromium-browser.list

This uses the dpkg data and avoids a large /.git directory and workflow.