Actually this issue is dealt with here: https://askubuntu.com/questions/879495/update-after-purging-intel-01-org-driver-removed-gnome-software/879516
There is a problem with Ubuntu Xenial packages I believe, and could quite possibly be the cause of your issue. I don't think technically this is an elementary OS issue, but an issue affecting all Ubuntu 16.4 ...
I'm not sure if Sean Patterson's answer is enough to prevent it from being updated by the AppCenter in the future.
Hence, I'd also mark the package as held back using ‘apt-mark’ command in terminal with root privileges.
For example, making your package say Firefox unavailable for install, up-gradation or uninstall, type:
apt-mark hold firefox
To make ...
Install Synaptic Package Manager
sudo apt install synaptic
launch the program and search for firefox
select it, then select Package at the top
select Force Version...
from the pull down select the version you want
If you're comfortable with the command line, you can use the dpkg command to install a .deb package.
The format would be sudo dpkg -i name_of_package.deb
then, sudo apt-get install -f. the dpkg -i command may leaves the package status in an "unconfigured" state due to missing dependencies.
sudo apt-get install -f will try to fix it downloading and ...
in the shell you can remove a package with the command:
sudo apt remove packagename
after that I would suggest to run the following command to remove unused dependencies:
sudo apt autoremove
You can also remove package from the app center, just select the package and click uninstall.
If you installed a third party kernel and then removed the source this could happen
It's not ElementaryOS's problem (because EOS didn't had that kernel version - Release Candidate) but I may help you
You can remove the package even if it's broken with the command:
$ sudo dpkg --remove --force-remove-reinstreq <package>
And lastly, it's sudo apt ...
The question is a bit vague, but How to install MySQL on Ubuntu 16.04 will take you through the steps, just make sure to update (sudo apt update && sudo apt upgrade) your system before installing mysql.
I also recomend MySQL Workbench, it just makes your life easier...
I believe you can only be sure if you review each package and it's dependency (time consuming)
I do autoremove every time and nothing bad happened yet, however people did complain in the past that it removed necessary packages