OK.

So, if I got PS3, I'd need to get soemthing like the HM-31. Using a DVI/HDMI cable, I'd hook the OPPO 971H to the HM-31. I'd use an HDMI cable to hook up the PS3 to the HM-31. Then I'd use a 2nd DVI/HDMI cable to connect the HM-31 to the TV. That, in turn, would get me 1080i video.

Is the difference between 1080i and 1080p significant?

Any recommendations on a 1080p TV in the $1500 range?

You've got it, all the way through. HDMI/DVI from the 971H to the switch, HDMI fron the PS3 to the switch, HDMI/DVI from the switch to the TV, and you're all set.
It depends on a lot of factors. Screen size, viewing distance, display type, and even source material will all be significant factors. Personally, I wouldn't pursue a display upgrade solely to have 1080p. If there were other reasons to make an upgrade, that'd be one thing, but just for the sake of the "p" alone I'm not sure it's going to be a great deal. Remember, also, that the 971H's maximum output resolution is 1080i. OPPO didn't start supporting 1080p until the 981HD.
Others are probably better informed on the current TV offerings, but screen size range will certainly be necessary to help with that. My initial guess is that your current CRT would be hard to beat in its size range even though it can't do 1080p, and a larger display that offered 1080p with better performance than the CRT would likely land outside that $1500 price range. I could be wrong, though - like I said, I haven't been keeping a real close eye on the current offerings.

OK - thanks for all your help. It really is appreciated.

In the $1500 HDTV range, the Samsung LCD panels and the Panasonic Plasmas seem to be getting the highest ratings for their respective technologies. Sony's are popular for the name but not always the highest rated. You would probably looking at a screen size somewhere between 37" and 46". Look at some of the reviews online at like CNET or Ecoustics.

You'll end up getting a bigger screen if you buy last year's model instead of a smaller screen featuring all the bells and whistles for the same price. Get the right screen for your room, not just for the price.

also.... you can find DLPs in the 50" range from Samsung that will fall in the $1200-$1500 range.
They are also energy savers.
you will find that on CNET also....
Happy hunting/playing....