I noticed that JavaCard 3.0 may have the ability to use HTTPS from the Oracle website (oracle.com/technetwork/articles/javase/javacard3-142122.html).
Are there any ways to create HTTPS connections to a normal Internet website ?
A1: There are no JavaCard Connected (which describes such option) devices publicly available.
A2: Classic JavaCard does not specify/allow any kind of connections.
Basically with Java Card Classic you are limited to the APDU interface. This interface has been specified in the Java Card API and the ISO/IEC 7816-4 standard.
It is of course possible to channel any kind of protocol through an APDU interface, but you would have to program it yourself. Furthermore, you would have to do so on the terminal side as well, because Java doesn't know anything about TCP/IP, name resolution etc. As Java Card environments are very limited, it would be tricky to create something that resembles an HTTP client.
There have been demonstrations that implemented a tiny web server on a Java Card. Those obviously also require some kind of proxy on the terminal side.
The Connected Edition - if you can find it anywhere - uses the same idea; it implements a web-server for e.g. authentication. It doesn't provide a client to my knowledge.