Cannot reinstall SQL Server 2008 R2 after downgrade from Windows 10 to Windows 7
A few more suggestions—adding as an answer to accept more characters so I can include more. . .

BE SURE TO RUN EVERYTHING AS ADMINISTRATOR


Command prompt (as administrator) run SFC /SCANNOW - Once this completed, check the log file it points to in the window for other errors and post back that as an edit to your question, I think it's called CBS.log, so just look for error with the date and time around the time you run once it's complete
https://support.microsoft.com/en-us/mats/program_install_and_uninstall
You may want to ensure this is installed (or uninstall re-install)
https://www.microsoft.com/en-us/download/details.aspx?id=21254
If you have Visual Studio installed, you may want to consider uninstalling and reinstalling it as well
If there are any other prerequisite components installed on this system for SQL Server 2008 R2 Express, you may want to consider uninstalling and reinstalling those as well - see notes below


Note, sometimes with the SQL Server and Visual Studio and other components, software, etc. as such, you have to install them in the order based on their version or things in the Windows registry, DLL component, etc. just get out of wack.  

For example, if you have some component (or software) of the 2005 and 2008 version you need, it's best to install the oldest first (2005) and then once that is complete install the newest (2008).  I've seen this too many times where someone installs some old component or software as such on top of newer software, etc. that's already installed then the system start having strange problems.

If all else fails, you can backup your important data, and then do a fresh wipe of your HD and install a fresh copy of Windows. Microsoft has no way of emulating all software on everyone computers in the whole world, so perhaps when you upgraded and then downgraded, something just got corrupt and there was something on your system that they didn't test with, etc.  I typically always do fresh system upgrades rather than in-place upgrades but I'm in a business environment mainly too.