I know this whole aspect ratio thing can be confusing. Nonetheless, here are a couple of articles that can help to explain it:
To try to put this into perspective, my screen, (which you have seen), is a 2.35:1 screen, 120" diagonal. The seating is 11.5' from the screen. When we watched King Kong and FotR, that was 2.35:1 content shown on a 2.35:1 screen. The entire screen was filled with the image. The "widescreen" image was very wide and immersive. We then watched Avatar, and the screen was filled top to bottom, but there were black bars "pillarboxing" the image. This was 16:9 content on a 2.35:1 screen. It was still very immesive, but not the same as it would have been if it had been shown on a true 16:9 screen. A true 16:9 screen would have displayed this content in a much *bigger* format, not as wide, but taller and with an overall "bigger" presentation. 16:9 content on a 16:9 screen is the ideal way to display 16:9 content... but 2.35:1 on a 16:9 screen will have letterbox black bars above and below the image, as you now have with your current TV.
Bottom line, content is always best when displayed in it's original aspect ration and on a screen that IS that aspect ratio. There is no single aspect ratio screen that will be optimal for all different aspect ratio content. Therefore, if you're displaying multiple aspect ratio content, it makes the most sense to choose a screen that is the native aspect ratio of your "most viewed" content. For you, that would seem to be a 2.35:1 screen.
If you decide that you really want to go with a projector and screen, my recommendation would be to go with the largest 2.35:1 screen you can accommodate, which would probably be a 70" - 72" diagonal 2.35:1 screen. This would give you the biggest image for the content you view the most. A 72", 2.35:1 screen at a viewing distance of 7' or 8' would be hella immersive, and the PQ of a "widescreen" image that size, using one of these high brightness/high CR projectors, would be absolutely stunning!!!! Optimize the content you watch the most! Then, you can decide if you want to use the projector/screen for your other, less-viewed content, (16:9 4:3, etc.), or if it's more appropriate to view that content on your existing 16:9 TV.
The other significant issue for your situation is the dropped "height" of the screen. If you want to do this, we would need to find a screen supplier who can do a custom screen with the ability to add some extra "drop." This would allow the screen to be lowered to the proper height for your viewing position. This will add to the cost of the screen, but it can certainly be done.
One last comment... Please don't be concerned with the image bleeding off the top and bottom of a 2.35:1 screen. What "bleeds" off the top and bottom of the screen with a 2.35:1 screen is the letterbox black bars. It is not a problem at all to have these black bars bleed off the screen because they're *BLACK!!!* In a darkened room, you won't see them at all... because they're BLACK! With a projector with a CR of 500,000K:1, the black bars are so black, they're not at all visible. They're simply not a problem.
Bottom line, this is a doable solution. It just depends on how much $$$ and FAF, (Family Acceptance Factor), you want to expend. Only you can decide those factors.