I would like to ask if there is any list of pre-installed applications in Ubuntu which are safe to remove.
Once, I tried to uninstall unnecessary applications from Software center, but it created some bugs. For example, I lost some icons from System settings, and then I had to do sudo apt-get install ubuntu-desktop
to recover lost data.
I would like to remove unnecessary applications, but I don't want to break my system.
Ubuntu comes with a lot of GNOME applications pre-installed for giving the users an overall good desktop experience. Many of them have a deep system integration ... so it's not recommended to remove them - a good example is nautilus. But there are features that are quite safe to remove ... such as the controversially discussed online search integration that Canonical provides in Ubuntu.
As this is an aspect being more opinion based ... I give you a list of packages that I have removed :
Note :
Uninstalling unity-webapps-common removes the Amazon Search feature from the Unity desktop environment. In Ubuntu 15.10 I had to reinstall unity-webapps-service afterwards.
The reinstallation of the package unity-webapps-service wasn't necessary in Ubuntu 14.04.
You first need to define what considers "break my system". A system without a desktop is not broken as is.
Besides that: the safest way is what you did and to take a look at this by "category" from Ubuntu Software Center and the "installed" group.
Besides "accessories" any of these should be safe to remove and will not remove anything important since they all point to applications. Remove them 1 by 1 and do pay attention to the list of packages it is offering to delete just in case.
And when in doubt check the packages online for what is depending.
1 more comment: what you did was probably overkill; to get those icons back in system settings all you had to do is re-install that specific packages. This should have been enough to fix it: