How do I log out of a deleted Apple Account?

Let me be more optimistic and say not necessarily a brick.

Refer to How to remove Activation Lock - Apple Support for the various ways to remove an Activation Lock. Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. Check the requirements and steps at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not detail what currently constitutes proof other than it “must include the product serial number, IMEI or MEID.” Apple may typically require the original receipt* issued by Apple or an Apple Authorized Reseller. If Apple resets the device this will give you access to the device but won't restore access to the account used to lock it. There is nothing that can be done to expedite an Activation Lock request.

*Apple recommends you "Keep your receipt after you buy an Apple device" --> Keep your receipt after you buy an Apple device - Apple Support

If you cannot get the device unlocked then read this section from the very bottom of Apple Trade In - Apple :

"Does Apple offer recycling?

Yes. Apple Trade In lets you recycle any Apple device (including devices from Apple-owned brands) at any Apple Store and on Apple for free. That includes your batteries and old electronic products as well as free, on-demand packaging recycling for our commercial, education, and institutional customers. When we receive your device, it will be thoroughly inspected to determine if components can be recycled or reused. Whether recycled or reused, all activities relating to the processing of your device will be managed in an environmentally responsible way."

The account is already gone, that's why you cannot now use it to sign out of the device. Apple no longer has the sign-in information associated with it.

If Apple approves your reset request, the phone will be in factory condition.

Your account is already gone, but the device does not know that. The iPhone is signed into that account but Apple has already deleted the account from its servers. That's why when you try to sign out of the phone now you are getting an error message. The phone is contacting Apple to get confirmation of the sign-in information and Apple is replying, "What account? We have no information about that account. You must have entered the information incorrectly. Sorry, we can't sign you out of an account for which we have no information." In deleting the account before signing out of the device you basically cut yourself off from being able to sign of of the account on the device.

Factory condition means just that. No Apple ID, no information of any kind. You turn it on and it replies "Hello" and asks you to continue to do the set up, just like if you walked into a store and bought one.

You will not be able to sign out.

See How to delete your Apple Account - Apple Support

Probably, yes.

If you forget to sign out, you might not be able to use your device when your account is deleted.

You're welcome. :-)