On the BX I'd say to use a SATA controller, as even ignoring for a moment that it can only do UDMA-2, the PIIX is connected through the PCI bus, so no matter what you have that 133MB/s limitation, but at least the SATA card will be faster than 33MB/s.
IMHO ATA-SATA adapter make sense only on newer (or less old) stuff, where the southbridge has its own dedicated link with the NB, as it allows you to free up PCI bandwidth. Also the onboard IDE on these machines is most likely at least 100 so it doesn't cripple the SATA disk too much.

Yes, but only as long as the SATA card is broadly supported and has an onboard ROM with a BIOS extension for booting.

Do you really need more than 33MB/s on a PII / Win98 build? That's a pretty fast drive for those days. 😀 You could fill up your RAM in 4 seconds.

Personally, I would go with the adapter, since it's emulating the drive interface to something that era of computers can comprehend. I don't have any first-hand experience trying to get SATA adapters working on older PCs, but since that was kind of a Pentium 4 era thing, I would expect it to get dicey. Maybe I'm wrong. I'm sure people here do it everyday.

I feel like less complexity to your build is usually a better bet. IDE is probably safer. I have heard good things about promise cards though.

BX motherboard's on board controller will be limited to ~160GB without drive overlay software.
If your using a larger drive then that I'd probably go with a card over overlay software.
If your under the limit I'd go with an adapter just to keep it simple.

Performance wise I couldn't say which is better, but really both are pushing the motherboard to the limit anyway

Performance-wise the PCI SATA controller (if a decent one like Promise) will definitely be faster, but:
- latency is far more relevant than throughput for the feel of speed in general use, and the difference there will be tiny.
- any (non-JMicron) SSD wil so vastly outperform any period HDD in that that any differences between controllers are insignificant compared to the huge gulf vs even the fastest period HDD.
- PCI SATA controllers (or any other card with option ROM) seriously impacts boot time, whereas PATA-SATA adapters don't affect it at all.

Anything older than Win2k/XP, I just use adapters (not for lack of drivers, Promise SATA150 TX2 Plus and TX4 have perfectly good Win9x drivers), and newer it really depends on what I'm doing.