"Does this mean that if I find a motherboard with Apple Part No. 820-0991-B, that it will be a 333 MHz motherboard, or there something else involved in the equation?"
No, the motherboard part number indicates nothing about processor speed. An 820-0991-B motherboard could have any processor installed in it at this point. Back when the computer was new, if it were a Build-to-Order unit manufactured in mid-to-late-1998, processor speeds varied. The stock CPUs in these G3s shipped with a designated color-coded jumper block (233 MHz: red, 266 MHz: white, 300 MHz: black, and 333 MHz: green). One of the easiest ways to determine processor speed on an unmodified, bare motherboard would be to check the color of the jumper block.
"If I replace the 266 motherboard with one out of a 333, is that all that is necessary to make it a 333 machine?"
Yes, but check the jumper block color. Regardless of the processor speed, the incorrect jumper block can slow it down to the speed I've indicated above. Also, there's a power supply jumper (J28) on the motherboard, close to the back near the PCI card slots. Depending on whether you have the desktop or mini-tower model, the jumper may not be set correctly for your computer. For use in a desktop, the jumper should be set to "MAC." For use in a mini-tower, the jumper should be set to "PS/2." An incorrect setting will prevent the computer from powering up, and possibly give you the impression that the motherboard is defective.
The first two motherboards (820-0864-A and 820-0864-B) will have the ATI 3D Rage II +DVD graphics chip. The third revision (820-0991-A) has the ATI Rage Pro graphics chip, and the fourth revision (820-0991-B) has the ATI Rage Pro Turbo graphics chip. The support for master/slave configuration of dual drives on a single ATA channel is controlled solely by the installed ROM card and not by the motherboard version. The original ROM card ($77D.40F2) had this restriction, but the two subsequent revisions ($77D.45F1 and $77D.45F2) didn't. That's why some owners of late-1997 or very early-1998 beige G3s replaced the original ROM card with either of the two latter versions. I did that with one of mine - a 300 MHz desktop model. Concern about the onboard graphics chip was a non-issue, because I installed ATI Radeon 7000 graphics cards in all of my G3s.
If a beige G3 shipped with the DVD drive, it would also have the A/V personality card with the onboard MPEG decoder chip for watching DVD movies. The processor speed would be the 300 or 333 MHz version. The earliest versions of the "Apple DVD Player" program depended on a hardware-based decoding scheme, so a fast processor was needed and the display had to be connected to the onboard DB-15 monitor port for the setup to work. The stock SGRAM was 2 MBs, but could be upgraded to 4 or 6 MBs. By the late-90s, that was considered incredibly small. Having a Radeon 7000 graphics card with 32 MBs of DDR memory in my G3s, I didn't want to watch movies on a display connected to a slower onboard GPU. Running a modified version of the "Apple DVD Player" program that used software decoding, I was able to watch DVD movies on a display connected to the Radeon card. On the 266 MHz G3s, it could lag at times, so performance was definitely better with the 300 or 333 MHz processors.