I would recommend installing using OSGEO4W Network Installer, which is a package manager for open source GIS software. It gives your more visibility of the installation process and more flexibility to work out what may be going wrong. See What is OSGeo4W?
QGIS, like many open source GIS applications uses other open source components as part of its environment. These can all be bundled up in a standalone version from http://www.qgis.org/en/site/forusers/download.html. However there is another method using the OSGEO4W Network Installer, also available from the same download page.
As I understand it, OSGeo4W is an umbrella installation package that allows the compatible version of components to be installed within the same installation environment. It gives better visibility of what is installed, and allows individual components to be updated if need be.
As you have researched, if there is an old version of QGIS uninstall this first in the usual way, and it is important to delete the folder C:Users/Program Files/QGIS
Download OSGeo4W from http://www.qgis.org/en/site/forusers/download.html
Run the executable file and choose ‘Advanced install’ on the first window, then progress through the windows until the ‘Select Packages’ options.
You can check for any latent components on your system by opening each category one at a time and check that each line shows “Skip” next to the package.
This will show that GIS components have been successfully uninstalled. They will probably all be showing skip, unless you have other open source GIS software on your system. If you do and you don't want them click on skip until it changes to uninstall, or if you do want to keep it leave it as skip, or perhaps install the latest version?
In Desktop category click on Skip on the line
qgi-ltr: QGIS Desktop (long term release) - To add the latest LTR
Or Skip on the line
qgis: QGIS Desktop for the latest release
You can choose to install other components, or other GIS software for example if you use ECW files, like I do in my work, then you will need to open the category
Libs and click on Skip on the line
gdal-ecw: ECW Raster Plugin for GDAL.
(ECW is a non-open source component, and this plugin is not installed as standard through the OSGeo4W installation.)
If you are unsure I would recommend keeping it simple and just installing QGIS for now and come back and install other components as you need them, and as you run through the installation one of the screens is to install
Unmet Dependencies which are packages required by QGIS, but not selected. So you don't need to worry about working out what other components you need as OSGeo4W does this for you automatically.