I had a similar issue with the general "TeamExplorer/vs_teamExplorerCore" error and finally got Visual Studio 2015 Community Edition Update 2 to install with these steps on Windows 7 (some of which are outlined here and here.)
Since you said you have tried at least some steps in the first article, I will note that it wasn't until I combined steps (and added a couple) that this worked for me after multiple repeated failures. I had previously tried some steps individually without any luck.
Note: If you are only interested in trying the VS 2015 Community RTM installer for some reason (rather than all the steps outlined), I have listed a link to just that file at the bottom of this answer.
The steps I took, in order, were:
1.) Update Windows - Make certain Windows is up to date (as necessary, including reboots).
2.) Enable all four evaluations of symbolic links:
Check to see if all four evaluations are enabled. Run cmd.exe (Run as Administrator):
fsutil behavior query SymlinkEvaluation
All 4 evaluations should be enabled. If they aren't:
fsutil behavior set SymlinkEvaluation L2L:1 R2R:1 L2R:1 R2L:1
Once those 4 evaluations are set, clean up your old directories/registry keys, temporary files and package installation cache (below).
3.) Uninstall any entry for VS 2015 in the Control Panel - If present in the Add/Remove Programs section, uninstall any entry for VS 2015.
Note: I did not manually uninstall any of the dependencies that were installed during the failed installation (e.g. .NET x.x.x, IIS 10 Express, etc.).
4.) Clear out old VS 2015 files, folders and registry keys:
C:\Program Files (x86)\Microsoft Visual Studio 14.0
C:\Program Files\Microsoft Visual Studio 14.0
C:\users\user\Documents\Visual Studio 2015
C:\users\user\AppData\Roaming\Microsoft\VisualStudio\14.0
C:\users\user\AppData\Local\Microsoft\VisualStudio\14.0
C:\users\user\AppData\Local\Microsoft\VSCommon\14.0
Note: Of course the first two might be different depending on where you attempted to install VS 2015.
Clear your registry entries with:
REG DELETE HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\VisualStudio\14.0
REG DELETE HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\VisualStudio\14.0
REG DELETE HKEY_CURRENT_USER\Software\Microsoft\VisualStudio\14.0
REG DELETE HKEY_CURRENT_USER\Software\Microsoft\VisualStudio\14.0_Config
5.) Remove cached or temporary items - Completely remove all items in
C:\ProgramData\Package Cache
and
C:\Users\UserName\AppData\Local\Temp
Note: Temp you can be more selective with if you like since some items will be clearly unrelated to VS.
6.) Uninstall previous redistributables - Uninstall any previous redistributables you may have. I made it a particular point to uninstall all three copies of 2015 (RC, Update 1 and Update 2).
If you are unsure about uninstalling all three, I would suggest at least doing Update 2, even if you leave the other two alone (since I am assuming you are attempting to install VS 2015 Update 2).
I had previously manually removed:
c:\windows\System32\vcruntime140.dll
c:\windows\SysWOW64\vcruntime140.dll
but did not do this on the final attempt outlined here.
Note: Make sure these are not in use when you uninstall them. If you receive any complaints, stop any associated programs before continuing.
7.) Reboot the PC.
8.) Temporarily disable AntiVirus, if any - The brand I use has the ability to disable all items until a time of my choosing.
Note: I would suggest that if your program has a time limit on disabling items, you ensure that time limit is not reached while attempting to install VS 2015.
9.) Clean up any remaining registry entries with CCleaner, as needed - In my scenario, there were still some unused keys left over after uninstalling the last VS 2015 failed attempt.
10.) Reinstall VC++ redistributables manually - all VC++14 versions were reinstalled in my scenario but I would be especially concerned with Update 2 or whatever you are trying to install.
The redistributables I used for Update 2 were version 14.0.23918.
11.) Use the VS Studio RTM installer appropriate to your installation - Per the chosen answer here, there may be a problem with using the current installer for VS 2015 Update 1 or Update 2.
For my scenario particularly I:
Downloaded the Visual Studio 2015 Community Edition RTM ISO.
(SHA1:BAAD3CEBAB7A5834D8F78F7D02E4880C010F3BA9)
Copied the installer "vs_community.exe". For this scenario, I was not using the ISO contents, so I unmounted it.
Ran the RTM installer from a directory without spaces. I am unsure if this was strictly necessary, but a comment made elsewhere a long time ago seemed to indicate that this might be a potential problem.
Used the default installation directory (C:\Program Files (x86)\Microsoft Visual Studio 14). Previously, I had used a installation custom directory.
Had about 20-25GB free, since there is a lot to download and install. =P
If you need other VC 2015 RTM ISOs other than for Community, links can be found at this blog post.
If you trust strangers on the internet, this is a link for the VS 2015 Community RTM installer (only) I used, no 3.9GB ISO download required. =P
Regarding the installer, as I mentioned, I did not have to use the RTM ISO. The RTM installer pulls web updates just like the installer for Update 1 or Update 2.
Using the RTM installer I successfully installed VS 2015 Update 2 with all optional C++ selections, Python Tools for Visual Studio and Clang with Microsoft CodeGen via the web. I chose not to install the Web Development Tools, but I don't believe this made a difference since I had previously had the TeamExplorer Core error prior to successful installation with no options selected whatsoever.