Our new national ID cards (called 'citizen card') here in Portugal work like that.
They have a keypair and the private part is never transmitted out of if: the card itself can (when plugged in a reader) sign and encrypt data sent to it. So it's secure even when using a public (not trusted) machine; I mean, it can spy on your connection, but it can't clone the key.
The certificate can be used in browsers to login to websites that support it or any other applications that implement the right PKCS standard.
Yes it does. And it's pretty cheap too. I have an OpenPGP v2 smart card and an SPR-532 smart card reader. I will be writing up how I did it on my blog at https://grepular.com/blog/ probably next week. Need to find an hour or two to write the thing up.
I picked up an OpenPGP card and reader a few months ago and love the technology. My only beef is that the reliability of the software that talks to the smartcards is awful on the Mac.
Right now, it's definitely an approach to take if you are crypto nerd -- it's not ready for regular humans.
Not to my knowledge, but similar approaches have existed for a long time now e.g. RSA SecurID [0]. One day I'm going to set up my Yubikey to perform a similar purpose, but my motivation levels correlate with my paranoia :)
I have a yubikey myself. The smart card method is much better though. When using a smart card, it is impossible even for the owner (who knows the access pin) to read the actual key. The crypto operations involved in the authentication take place on the card it's self. The smart card is protected by a pin, and will wipe it's self if the pin is entered incorrectly too many times. You can even use a smart card reader with a hardware pin pad built in so neither soft nor hard keyloggers can intercept it.
The weak point is you still don't know what you're signing, who or what you're authenticating, etc. if you don't have a trusted display and do the entire transaction on trusted hardware. (your host may be compromised, and could show arbitrary data, proxy connections, etc.) Protecting the key does go a long way, but even with a secure passphrase input and smartcard key operations, there is risk.
The ideal, IMO, would be some secondary hardware with a limited, secure OS which lets you put specific limits on how the key is used, showing a hash or identifying details of each transaction on the secure display, prompting you to verify each. E.g. to send a payment, the amount and payee (and date/sequence number) are shown on the secure display to verify before authorizing.
