Account hacked, how to reset the security questions?

Option 1: Enable Two-Factor Authentication (2FA)

If you have access to a trusted iOS or macOS device (or can sign in on an iPhone): 

1. Go to your Settings (or System Settings) on an Apple device.
2. Tap your Name / Apple Account.
3. Navigate to Sign-in & Security.
4. Enable Two-Factor Authentication (2FA) using your own phone number.
5. Enabling 2FA effectively bypasses and replaces the old security questions system.

Option 2: Use Apple Account Recovery (iForgot)

If you cannot sign in through a trusted device:

1. Go to Apple iForgot.
2. Enter your Apple ID email address or phone number.
3. Follow the prompts for Account Recovery. Note that this may trigger a mandatory waiting period while Apple verifies your identity to protect your data. 

If this follows a common pattern, this involved a re-used password, or a phished password, and no two-factor authentication.

If so… That’s not your account anymore.

Your sole remaining option to try to wrest back control — this might work — is adding a low-limit virtual payment card. Then using that payment card to try to use Account Recovery to regain control.

That’s described here: iPhone, iPad or Apple Account Security Co… - Apple Community

The takeover of an Apple Account with a re-used (compromised, phished) password and no two-factor authentication is trivial.

You’re right, you can only delete an account into which you can successfully authenticate.

Unfortunately, this is a not an uncommon scenario for seldom-accessed; old accounts, not protected by 2FA.

It’s likely not so much that Apple is being unhelpful, but the way those accounts were created and secured was entirely by your self-created credentials (password).

i.e. You never appeared in-person before an “Apple Agent” and presented “2x forms of Gov’t-issued ID and 2x proofs of residence” as you might for driver’s licenses).

Access and control therefore effectively belongs to whoever holds the credentials … and unfortunately in this case … that’s now apparently someone else.

Wheelie Nick gave you all the correct answers.

But you might also be thinking ahead for the event that you never regain access to that account …

… as you definitely have some significant obstacles standing in your way.

Honestly, I'd be happy to just delete my account, since I haven't used it in years anyway. But even this seems impossible as long as I can't answer the security questions.

And all I have at hand is a super old iPad, which doesn't support anything and on which I can't setup 2FA.

Starting to think that it's a lost cause. Apple support hasn't been particularly helpful so far, but thanks both for your support 🙏

i.e. You never appeared in-person before an “Apple Agent” and presented “2x forms of Gov’t-issued ID and 2x proofs of residence” as you might for driver’s licenses).

True...but I would have hoped that owning the email address linked to this account would be a sufficient proof that I am the legitimate owner :-)

Anyway, at least there are no credit cards or important data on this account. Only frustrating to have been hacked / robbed