Consequences of setting up Face ID on a compromised iPhone

Nothing. Nobody has access to your FaceID data, not even you. It is stored in the Secure Enclave which has never been compromised and the data cannot be shared. That is why when you set up a new device, you have to set up your Face ID again as there is nothing to transfer to the new phone.

Even jailbroken the Secure Enclave, where biometric information, encryption keys, Keychain items, are stored has never been hacked.

Quad9 is a DNS server that is widely used and I would have no problem recommending it. It requires no software to be installed on your device and is free. All you need to do is add 9.9.9.9 to your IPv4 DNS Setting on your network and 2620:fe::fe for IPv6. That is the default setup and how Quad9 got their name. They have added different addresses that can be used to include other features like blacklisting known Malware sites.

I do NOT recommend installing any VPN. You should expect network issues and they do not provide any security to your device at all. In fact, it weakens security by adding another hop in your network traffic and the servers of those VPN providers have been breached before resulting in data compromise including passwords. And those are the legitimate ones, others are "sniffing" your network traffic across their servers and selling your data. That is how a "free" VPN provider produces income.

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If you did not jailbreak it, it has not been "compromised".

Why do you think it was?

I'm aware of that. It was more a question as to why the OP seems to believe the phone was compromised at all.