I am having the same problems. I told our IT department to stop upgrading operating systems for GIS users until we found a solution. (Waiting is not an option right? we only have a few more months for free Windows upgrades!)
I have ArcGIS working intermittently on 5 Windows 10 machines. Here is the kitchen sink approach I used to finally get one of them working normally. (if nothing else, maybe one of these might help) I will try it on the others to see if it is a repeatable fix.
My symptoms were the application failing to startup (very commonly without even showing the splash screen), the application starting up but all of the connections had red exclamation points, and multiple error messages when trying to start the application. The failures are not consistent across each machine. Some machines work fine, but the Find tool might not allow navigation or the color palette doesn't come up (requiring RGB values to be entered). Others require an uninstall/reinstall every three or four days
My steps
1. Installation
* Uninstalled and reinstalled to 10.3.1 (seemed to work with 10.2.2 also)
* If you can, set concurrent use licensing to advanced. I got it to work with lower categories after the initial installation
* After installation, rename (if it is still there) the default geodatabase to something like default_old.gdb.
I found it in c:\users\{user profile}\My Documents\ArcGIS. If you see a bunch of them (Default, Default1, etc), I
would try copying them into a backup directory if you are uncomfortable deleting them all.
* I also got rid of the ESRI profile (c:\users\{user profile}\AppData\Roaming\ESRI). It will recreate itself when
you start ArcMap. (again, rename it, if you are uncomfortable)
2. First run
* My first run, I ran the application from the user's profile as an administrator. It would not run the first time without elevated priviliges. Thereafter, it worked fine.
3. Miscellaneous
* I could not make direct connections at first. A little sleuthing showed that the database login for the generic "read only" user account had a default database set to an edit database it was not mapped to. As this account was made via geoprocessing tool, it did not occur to me to check it at first. This caused strange error messages when trying to make a direct connection.
In this era of application development, it is truly a shame that there is not better exception handling and reporting in their startup routine.
