The only way, I believe, to connect an iPad wirelessly to a projector is by connecting an AppleTV to the projector - and I don't think most projectors have the correct input for it.
You can use a number of apps to control the computer through the ipad. There are a few apps that are designed to replicate an interactive whiteboard. One I'm trying is Splashtop Whiteboard - it allows you to use the iPad to annotate over an image on the computer and then project it, or to even use their own flipboard as a background to writing. This would, in theory, allow you to do your work directly on the iPad and send it to the projector through the computer. I say in theory because we are having problems with it, both the app itself and then running it through a whiteboard projector - I've been working with a tech support rep at Splashtop to see if we can resolve it before I can decide whether to recommend to other teachers, as right now I can't - but other people apparently are using it succesfully.
Another app that has similar promise (and higher reviews!) is Doceri Remote. The app is free, but the desktop software is not - $50 (the splashtop app is $20), plus they strongly recommend their proprietary stylus. Because they sell the computer software and not the app, however, you can do a trial - 5 days, I believe, we're going to give it a try after winter break. You can also, I believe, record your presentation using Doceri.
For recording ... while Doceri promises it, it might be easier to use a Screencast software (such as Jing) to record it on the computer - if you're using one of these apps to control the computer and project out, anything on the computer will be recorded by Jing and then you can share it later.
