I haven't used it yet, since our school is primarily Windows, but one teacher plans on trying it with her Mac. Reflection is a fairly new app that allows you to mirror an iPad onto a Mac (which could then be connected to the projector).You will be able to use the Promethean software, which will not work for options 1 or 3. This won't show the iPad screen on the board, but will allow you to show the computer screen and control it from the iPad. Use the iPad to control the computer (with an app such as Remote Desktop).
The downside is that it's another device ($100) to connect, requires an additional power source, and since most school projectors don't have HDMI inputs, you'll likely need an HDMI-VGA connector to link the Apple TV to the projector. This allows you to "mirror" (duplicate so you see the same image on both screens) the iPad screen to the projector and thus to the board. While as of now the only way to go directly from the iPad to the projector is through a cable, you have several options to go through a second device. You can not go direcly from iPad to Promethean board (you need to go through the projector). Not directly, and not without some help, but there are ways to get that effect.
Can I wirelessly project what's on my iPad screen onto a Promethean board? I have read of connecting the iPad to the digital projector via iPad-HDMI cable, but I want to be able walk around the classroom, not tied to the projector.