CS6 has not been updated or supported for quite some time now. It works on the operating systems that Adobe said it did when they released it (when you originally bought a license for it). Adobe is not required to update old software for new operating systems that came out after they ended support. Adobe does support InDesign CC and that will work on Windows 10. You can still use your InDesign CS6, but it may have to be on an older version of Windows.
One thing to keep in mind, some apps may not work well after a system upgrade even if they normally would work on that operating system. I'm not sure if InDesign CS6 will work on Windows 10 (others can add their experience if they have tried running it there), but maybe uninstalling it and reinstalling could fix the problem?
I assume you did the free Windows 10 upgrade in place over your installation of Windows 7. From what I've seen that is not a very reliable upgrade. I would recommend, if you intend to use CS6, that you back up all of your data and then do a full reset of the operating system followed by a re-installation of CS6.
Settings>Update & Security>Recovery>Reset this PC (from here can choose to keep all of your data, but I still suggest a back up to be sure)
It still won't be supported but I'm running it here (only occasionally) and I haven't seen any issues at all.
It seems I was able to update my non-Cloud version InDesign to CS8.1 thanks to a free—and quite limited—subscription to the Creative Cloud, as described here. I am contemplating moving that machine to Win10, and am grateful for Bob's cautious experience concerning the free, in-place OS upgrade. But if I were in the OP's shoes I might try the free CC subscription first in case the IDCS8.1 update would allow him to continue working and schedule the full-bore OS upgrade at a more convenient time.
While I agree that Adobe is not obligated to update software indefinitely, including for new OSes, I know for a fact that InDesign CS6 v.8.1 works fine in Win10. My point is that I should have access to the version of my licensed software that works, and Adobe does not make it available for some reason.
I have the same license installed on my desktop and then on my laptop (for when I am traveling), as allowed by the license. Both computers are running Win10. My desktop has InDesign CS6 v8.1 and it works fine. My laptop, for some reason, has CS6 v8.02 (it must have missed that update somewhere along the line), which does NOT work after upgrading to Win10. When I check for updates using the Adobe Update Manager, no update is available for v8.02, even though I know that it exists. Adobe does not make v8.1 available on their update website either. My request is for them to honor my license and provide me with a way to acquire v8.1 on my laptop.
8.1 contains features only available to those with a CC subscription. Pay for it and you can have it.
But even at 8.1 CS6 is unsupported.
According to this post Adobe InDesign CS6 8.1.0 update missing from downloads page the 8.1 update was originally only for Creative Cloud members, which is why you weren't seeing it. I does look like that was mainly a bug fix release, so I don't know why Adobe would only have made that for CC members and not perpetual licenses. Someone in that post said it was available for them so I'm not sure what the story is.