Precisely, can I remove smart card from card reader once I successfully started a SSL/TLS session? Will website still 'see' my client certificate (public key) after I removed card from the reader and navigated on the website? Will I still be able to communicate with website in the same SSL/TLS session?
Precisely, can I remove smart card from card reader once I successfully started a SSL/TLS session?
Yes. I think that is the most likely scenario. The client cert is used to establish the initial connection and negotiate a SESSION KEY. This is the initial session negotiation. After that, you can theoretically RENEGOTIATE any time the server (and sometimes the client, too) feels like it. But that depends on the specific web app. However it would be out of the ordinary, and I don't think it's very likely.
But just try it yourself and/or ask the admins of the web app.