the powermid should work fine to extend the signals you are already sending. I think the powermid can work with just about anything since it does not need to actually understand the signal, just repeat it. The halx can also send IR via RF directly to the 2nd powermid, I believe.
As for whether the halx would be able to learn from your particular brand of projector I could not say. It all depends on the frequency used and the modulation protocol. Heh heh, can you tell I really have no idea what I'm talking about?
If you can tell Rebel the brand and model number he might have some idea about the likelihood of it halx being able to work with it. Obviously it is not possible for it to work with everything.