Only CRTs can display interlaced signals. All digital displays (plasma, LCD, LCD projectors, etc...) are progressive. As mentioned above, a 720p projector always outputs 720p, just as a 1080p plasma always displays 1080p. The is no such thing as a 480i or 1080i digital display. Of course these devices can accept interlaced signals-- standard def TV (480i), and high-def TV (1080i).
However in these cases, the display has to de-interlace the signal to display it progressively. The quality of de-interlacing can be variable. So perhaps 1080i on a display that does a relatively poor job of de-interlacing might look noticeably worse than a 720p source on the same display.
With most newer displays, the consensus seems to be that 720p and 1080i sources look very similar.
So to the OP, first of all, your processor should be set to output 720p, otherwise it will be sending a signal that your PJ needs to process additionally.
720p sources (some HD channels) need no processing whatsoever to display on your PJ. These sources should be sent 720p native from the cable box, pass through your processor without being changed, to your PJ which will display them natively.
480p from standard DVD doesn't have to be de-interlaced, but needs to be scaled to 720p for your PJ. You need to figure out which of your player, processor, or PJ does this better.
As to 1080i from your cable box, this needs to be both de-interlaced and scaled down to 720p for display by your PJ. Most likely either your processor or PJ will do this better than the cable box, and you need to figure which one actually results in a better picture.
If your PJ is significantly newer than your processor, it is possible that it does all of these things better, and you might be able to ditch your processor.
Hope this helps
dave