[EDIT: you may want to scroll to the bottom for the nuclear option which I wish I'd discovered earlier!]
I've downgraded two computers from Windows 10 to 7, both with VS2015 on them. One worked and the other had its VS2015 break horribly. I also deleted the VS2015 directory and registry data that mentioned VS2015 and all sorts, and I had even more problems than you describe :) I think the difference is that I installed some new things (node.js and TypeScript) under Windows 10, and they inserted things into VS2015 which were no longer properly installed after the downgrade.
Here are my discoveries in a hopefully useful order:
When it hangs, what it's actually doing is trying to display this prompt asking you to supply the path to an installer it couldn't find.
You usually get to see these prompts if you 'Run as Administrator' the (main) installer (rather than letting it obtain Administrator privileges itself). Process Explorer helps shed light on this: if the main installer's window is associated with the child process, then the dialogues are visible. If the root process, they're not.
The prompt relates to old versions of packages that the installer wants to uninstall prior to installing a new one, and for some reason the installer doesn't know how to re-download those packages. It is usually looking under C:\ProgramData\Package Cache for them.
If you aren't seeing the dialogue, you can view logs in C:\Users\<you>\AppData\Local\Temp. Use the 'Details' view and sort this folder by date modified, to help you find the right one. Each sub-installer tends to make a new file so you may have to poke around a bit. Changing dates or file sizes also tell you that it's doing something.
Messages like this tend to be associated with the attempt to show a dialogue:
MSI (s) (24:D8) [13:00:25:033]: SOURCEMGMT: Trying source C:\ProgramData\Package Cache\{388D7468-1CCA-40C8-9F08-4C20E972E922}v14.0.25123\packages\BuildTools_MSBuildResMsi_amd64\.
MSI (s) (24:D8) [13:00:25:033]: Note: 1: 2203 2: C:\ProgramData\Package Cache\{388D7468-1CCA-40C8-9F08-4C20E972E922}v14.0.25123\packages\BuildTools_MSBuildResMsi_amd64\BuildTools_MSBuildResMsi.msi 3: -2147287037
MSI (s) (24:D8) [13:00:25:033]: SOURCEMGMT: Source is invalid due to missing/inaccessible package.
So, what to do about these missing packages?
If you have another computer with a working Visual Studio 2015 installation, you can copy the entirety of Package Cache from there to your bad computer (no need to merge folders that already exist) and that will hopefully catch many of them.
For the rest, I tried a few third-party uninstallers, and Revo Uninstaller helped. Go into its settings and enable 'Show System Components'. Then, whenever you identify a problematic package, you'll usually be able to find it in Revo Uninstaller. If you uninstall it, you'll see the usual prompt (cancel it), and after it fails, Revo's 'Moderate' registry cleanup option does the trick ('Safe' didn't). If you use the Pro version then you can multi-select and use the 'Quick Uninstall' batch option, which isn't quite automated - yes, I'm afraid it will be tedious - but with some patience, you can get through everything.
Identifying the problematic packages is still a manual process, but the good news is that you can cancel many of the prompts during a run of the installer without it aborting, so you can collect a lot of package names as you go. You can also look for patterns in version numbers; for me, a lot of them were 14.0.15123` or something like that.
For me, just one package didn't show up in Revo Uninstaller, and I searched the registry myself and deleted some occurrences. I'm not sure if that or the subsequent install of the new package was what did it, but even that one went away.
So, after all that, I'm pleased to report that Visual Studio 2015 has got through the 'Repair' operation with no further errors. I still need to reboot before I can see if it runs without a hitch...
[EDIT: Nuclear option follows]
It didn't. Many of the standard windows (code editor and error list to name a couple) failed to load with the error "An item with the same key has already been added". On the plus side, at least Visual Studio didn't suddenly exit moments after opening. So that's an improvement and 2-3 days well spent :)
After that:
I found this answer and tried the TotalUninstaller linked therein.
Then I deleted more of the installation where that uninstaller reported it couldn't delete a non-empty directory.
Then I was going to reinstall from scratch, but would you believe it - it gave me Modify/Repair/Uninstall options! So I ran the Uninstall. (Maybe Repair would have worked, who knows?)
Then I reinstalled from scratch.
There was just one more failed uninstall of an old thing (Microsoft Web Deploy), but it didn't even report a warning at the end because of it, and now my Visual Studio finally seems to be intact - fingers crossed! (I may never test that web deploy thing anyway - I mainly use it with Unity. Of course I had to install the Unity tools again.)
So yeah. I have no idea if this works by itself or if you also have to do all the other stuff. If you try it and find out, let us know.
