WyomingKnott :
Brilliant. A convenient, widely-used two-factor authentication device.
In response to both 1 and 2:
FIDO U2F is a challenge-response system.
And no, the service you connect to does not have a symmetric key, it is based on asymmetric public-private key cryptography, the only record of the private key is held by you, on your YubiKey.
Every service gets it's own key pair, so no 2 services have any knowledge of even the existence of another service, and the key pair is the only identifying feature, the service has no way of knowing whether you are using a particular key, or multiple keys.
The Yubikey is programmable, so you can control almost every aspect of it, including what services it has.
In terms of trust, YubiCo is one of the largest, and longest lived companies in the physical second-factor business, and is a big pusher for open-source security solutions.
I am speaking, of course, of the physical keys pictured, which their study named as being the most secure, as well as most convenient 2FA system, not about the duo push, or sms based systems, both are, imho, woefully insecure systems.
I too, am highly paranoid, and use a non-internet capable device for important password storage (though I do use the more convenient keepass for low-security passwords such as forums, that I use often. I have also been using YubiKeys as my 2FA system since before they helped design FIDO U2F, since I was introduced to them in the cryptocurrency world.