How to find which app has taken my F10 and F11 keys

You may find this discussion at SuperUser.com to be of interest: windows - Find out what program is using a hotkey - Super User. This is a much harder question to answer than to ask, it seems. Good one!
HTH,
--Ed--

PS: looks like using the Spy++ program that comes with Visual Studio will be your best bet, but there are at least two good other suggestions in this thread.

If it doesn't mention it, Process Explorer itself has a target which you can drag over a window to identify its process.
What program is displaying this window? - Ask Leo!

How is that going to help identify what program has taken over the F10/F11 keys?

Supports Ed's comment and URL content in post #2 - assuming you read that, and it's actually meaningful.

New program: Ethervane ActiveHotkeys (freeware) - DonationCoder.com

Please note that Windows does not allow for detecting which application has registered a particular shortcut, so that feaure is not included.

That was then. Have things changed? Hmm...

Some (older) possibly interesting programs that list assigned shortcuts (but don't solve the problem- see quote above):
https://www.nirsoft.net/utils/hot_keys_list.html
Download Ethervane ActiveHotkeys (portable) v1.3.1b6 (freeware) - AfterDawn: Software downloads
Shortcuts Map - manage hotkeys assigned to Windows shortcuts

Good pointers, @dalchina. I'm bookmarking several of them. I went to Nirsoft/Nirlauncher first, but didn't find the hot_keys_list item. There's so much to dig into under that umbrella now, it's often hard to find stuff. Thanks for reminding me about this, too.


--Ed--

I suggest using Clean Boot:

How to Perform a Clean Boot in Windows 10 to Troubleshoot Software Conflicts