!. It most certainly can stay. I would suggest you leave SU2014 at least until you are fully transitioned to SU2016.
No. they install and operate as separate programs.
Some of the customizations can be imported to SU2016. Shortcuts primarily. You can also use custom materials, styles and components libraries.
You should install any plugins/extensions fresh from their sources (do not copy them from the old install) and expect to manually set up toolbars that you may have customized in SU2014.
One thing I’ve found useful when setting up a new version of SketchUp is to make a screen shot of the older one and import that as an image in the new one. That allows me to see which toolbars I placed where and saves me having to go back and forth between versions.
You can install both at once. You can also export shortcuts from 2014 and import to 2016.
AFAIK you can’t transfer toolbar customizations from one version to another.
If your plugins come from SketchUcation plugin store, you can manage them via the SketchUcation Plugins and Extension managers. Recommended NOT just to copy from older to new Plugins folder.
Extension Warehouse offers similar facility, but I haven’t used it.
Download again latest versions of plugins - where possible in .rbz format and reinstall if not from either Plugin/Extension store.
Most but not all plugins that worked in 2014 will work in 2016. Many older ones will break - Ruby version changed from 1.8 to 2 in SU 2014.
2016 will install beside 2014, so no need to uninstall.
The interaction you will most likely see will be around “auto-opening” .skp files by double-clicking on them. The version that is used will depend on your operating system (you should update your profile, so we all know).
There is a process for migrating some of your presets/extensions, but the recommended process is to set them manually (especially in the case of jumping forward two versions).
… and Surprise, surprise, @DaveR beat me by a click!