I just installed BOINC manager on a new Windows 10 machine (BOINC Manager version 7.6.22 (x64). I changed my Windows firewall settings to allow BOIN.exe and BOINCMgr.exe to get through. However, it fails to retrieve current status, and complains about the failure every 3 minutes. Are there some settings I need to change?

What version and "As user" or "As service"?

No trouble here and only had to set an Incoming rule so I could remote in onto the client i.e. on the machine no rule was needed at all.

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Coelum Non Animum Mutant, Qui Trans Mare Currunt

I changed my Windows firewall settings to allow BOIN.exe and BOINCMgr.exe to get through.

I don't know how Windows firewall works but if those literally are the names you have in it then you have mistyped boinc.exe .

BOINC manager is 7.6.22 (x64).Windows is Windows 10 Home, v 1511, build 10586.494.

I ran BOINC manager by right-clicking on the icon and selecting "Run as administrator." Not sure what you mean by "as user" or "as a service."

BOIN was a typo - I selected the executable in Windows Explorer.

I ran BOINC manager by right-clicking on the icon and selecting "Run as administrator." Not sure what you mean by "as user" or "as a service."

http://boinc.berkeley.edu/wiki/Installing_BOINC#Install_options
While installing, did you let it show the "advanced" options and possibly check the "service install" box?

There might be an update available for you Windows...
http://www.computerworld.com/article/3102899/microsoft-windows/microsoft-delivers-windows-10s-1607-upgrade.html
https://support.microsoft.com/en-us/help/12387/windows-10-update-history

Before fighting with firewalls best to check Task Manager that boinc.exe is running.

I didn't use any advanced options. BOINC manager shows up as a running app. BOINC system tray for Windows is a background app. They are both shown as running.

BOINC manager is NOT BOINC client (boinc.exe). They're 2 separate parts of BOINC.
It's possible to have the manager running, but the important bit, the client (boinc.exe) NOT running.

I ran BOINC.exe. It failed while "Creating new client state file"

You shouldn't have to need to run boinc.exe manually, just let Manager do it.
Don't run Manager as administrator. It isn't needed and some file ownership/permissions might get set to admin only.

Check the permissions of BOINC's data directory. The default location is C:\ProgramData\BOINC\ . The directory is hidden, copy-paste the directory in Explorer's path bar. See if you can create files in that directory. If you can't try re-installing BOINC to fix the permissions.

I can create files in Program Files\BOINC. I tried running the BOINC installer again, and did a Repair. Still not running.

Wrong directory.

If you can create files in Program Files then either you elevated yourself to administrator (or disabled UAC) or the filesystem permissions are totally bonkers. Refrain from elevating, the client isn't doing that either.

edit: Once you have found the right directory and if you can create files there too then post here the contents of stdoutdae.txt and stderrdae.txt .

Both files are empty. Size 0, no contents.

Maybe you have 2 installs, but those 2 files being empty (and if not also same name, .old suffixed files existing), indicates the core client never ran. Date stamp of creation / modification gives further pointer if so.

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Coelum Non Animum Mutant, Qui Trans Mare Currunt

Check that there isn't anything BOINC in VirtualStore. C:\Users\User_name\AppData\Local\VirtualStore and HKEY_CURRENT_USER\Software\Classes\VirtualStore .

If there wasn't anything or removing it didn't help then try completely wiping BOINC using these instructions and then re-installing with default settings.