You can already get a web server that runs on your SIM card called a Smartcard web server (SCWS) if you have a GSM phone. It works in a part called SIM application software, which is a really cool area of your SIM card to run services. The web server is part of the OMA (Open Mobile Alliance) specification, so most (if not all) of the latest SIM cards support it. The web server can serve static and dynamic web pages, and you can also run some Java applets .
The problem is that it was intended for your mobile operator to serve the web pages on your phone, and not so that someone could remotely connect to your SIM card to view the content. For example, here in the UK, O2 uses this technology to provide a good interface for additional o2 features on my phone. I can choose traffic, football, etc. from the HTML menu, and then, when I expand through a small site running on my SIM card, I get a link that takes me to a deep-link site on the o2 main site.
I think there is a way to allow SCWS to be public, because in 2005 a press release was issued by Axalto (now Gemalto), where they stated that they have a SIM card that runs a personal blog for the user, and the user can take a picture on your phone, press the button to decide to publish it on your blog. Then his friends can have immediate access to him from a PC or mobile phone.
If you are working on remote viewing of SCWS, then the next problem is how to write files to your SIM card. Traditionally, you will need a kind of SIM card reader / writer that uses the ADPU protocol to talk to the card. However, the mobile network that issued your SIM card has remote access to the SCWS files on your SIM card, issuing commands over the air to your SIM card. This means that there is a way to remotely update files, and after a little copy, I think that security is based on a pre-shared key. Once you have the pre-shared key, you can send your own phone using an SMS message labeled Over Air Air, which will reprogram the SIM card to store new files for your web server. I can tell you how to get your pre-shared key if you send me a direct message.
So in short:
Is this technology currently available? - Sort. You can start the web server on your SIM card, and if you can open it for your public IP address, you can start the web server directly from the SIM card to the public.
How widely is it currently used? Operators use this quite a lot all over the world to serve you a little better value added on your phone. I have never heard of anyone other than Gemalto, using this to display public pages.
When is it going to benefit from classic remote paid web servers ? This is a difficult question. If you can get SCWS to work for open access, then the next step is an easy way to publish to a SIM card. When there is an easy way to publish on a SIM card, how do you get some kind of DNS system that works when a dynamic IP address from your network is issued to your phone.
What hardware (type of cell phone) will we need to use? . Since this is done on a SIM card and not on a phone, this is completely independent hardware, however SCWS is only available on later SIM cards (not CDMA phones).
More information here: http://smartcardwebserver.com/ and a good overview of how it works with Gemalto . Probably not bad to read the Resources section of the SCWS SIMalliance Working Group