Can I burn a CD with 2352 bytes/sector and not the error correction codes to get 800MB on a CD without overburning?
The PC CD-ROM format only supports 2048 bytes of data in each sector. This is a standard and can't be changed.

If you try to burn 2352 bytes of data in each sector, it will no longer be recognized as a PC CD-ROM.

The Audio CD format, for example, uses 2352 bytes/sector.
In terms of data you can fit 800 MB of music on a 700 MB CD without overburn.
This is why when talking about burning Audio CDs you talk about minutes and not actual data. It avoids confusing people.

I downloaded a few BINs just last night, and out of habit converted them to ISO and deleted the BINs. So... what does it matter that I essentially pruned the error correction data? Does that mean the CD won't work, or just that it will have problems?
Don't worry, the CD will work fine.
When converting BIN files to ISO, you lose the error correction codes, but the actual data remains intact.
Without the error correction codes (ISO format) you can still burn the CD with no problems.
However, when burning at high speeds, the ISO format won't be as reliable as the BIN format.
In other words the probability of burning coasters is higher when burning ISO files at high speeds.

That all said, how can I make a BIN on the fly from CD? I love how WinISO has the option to make an ISO on the fly from a CD. It has an ISO>BIN converter but I suppose that doesn't matter, does it? cuz the error correction codes are already gone.
When converting ISO to BIN the error correction codes won't be generated because they are already gone.
The ISO files will just change to comply with the BIN format (raw image), but no error correction codes will be added.
It's similar with MP3 and WAV files. When you convert MP3 to WAV you won't get a better quality audio file. Just the same MP3 with the same quality, but in a different format.
BlindWrite, CDRWin and Alcohol 120% can create BIN/CUE files from the CD.