First, on the setting two devices to "Master"...:
Setting jumper(s) to the Master or Slave position is relevant only to the particular IDE port they are connected to. Any IDE port MUST have a Master device to function; if there is a second device on the shared port / cable, then it must be the Slave. These designations only establish the unique identities of the devices sharing the IDE channel. They do NOT have any meaning for the machine as a whole - there is no such thing as a "Master Drive" for the whole computer. Most computers DO have one Boot Drive that contains the OS (some can have more than one), and many years ago by default this was the Master device on the first IDE channel. But that convention is long gone, replaced by the ability to specify in BIOS Setup exactly which device is the boot device - in fact, now one specifies a priority sequence of potential boot devices. So, having Master devices on more than one IDE channel is NOT confusing - in fact, it is necessary for the IDE channels to work.
Basically, the adapter you are using is supposed to use a SATA port to create an IDE port. On the IDE side (where you connect your IDE device like the Zip drive), the cabling and the devices need to be set up as normal IDE devices. In most cases, the adapter only supports ONE attached device, however, and that usually means that it must have its jumper set to be the Master. Some adapters may be different, which is why I suggested checking for instructions on this point.
On the SATA side, where the adapter connects to your mobo SATA port, the adapter/IDE device combo appears to be a SATA device as far as the mobo is concerned. Thus the OS needs a driver to know how to deal with the device on this port. Win Vista and Win 7 have such device drivers "built in". But Win XP does not, so the driver must be installed.
There are TWO ways that a device driver can be installed in Win XP. The straightforward way is to install it after Win XP is up and running. After this is done, the driver is located on your boot drive, and can be loaded from there by Win XP AFTER it has done most of its normal loading process. So it must be located on a device that Win XP can already use due to its own "built-in" drivers, because the device can't be used until the driver is loaded. The driver you are installing can be located just about anywhere because you can tell Win XP where to look for it (that means you need to know where it is). Generally you do this from Device Manager by right-clicking on the device in the list and choosing to install or update the driver for it. But what if you don't have the device in the list? This can happen if there never was such a device earlier so that Win XP could find it, and this is probably your case - up until you tried connecting the adapter and Zip drive, there was NO SATA device in the machine. So what to do? Make sure the adapter and Zip drive are attached with power supplied to it/them. Then make sure in BIOS that the SATA port is Enabled and that the Zip drive is detected properly. Then let Win XP boot, and go into Device Manager. In there RIGHT-click on the top level and choose to Scan for Hardware Changes. This will force it to go through all the connected hardware devices and find the SATA device that did not exist before. Then you can choose it and use the option to update or install the SATA device driver. (It is even possible that the Scan process will automatically search for and find the SATA driver on your C: drive.) If it asks you for the location of the driver, make sure your mobo CD is in the optical drive and direct Win XP to look at the right subdirectory of that disk to find the driver required. According to BioStar your system has the ICH7 Southbridge, so that's the driver you need. (However, sometimes the driver appears to be in a subdirectory above this, because that's where a file is located to direct Win XP exactly which driver to use.) Let Win XP install that SATA driver, then back out of Device Manager and reboot to ensure it's complete. Now see if the Zip drive appears in My Computer.
The second major way to install a driver in Win XP is what BioStar's Tech Support told you they thought is the only way, and they are wrong there. This method is specifically for installing a device driver as a "built-in" component of Win XP so that the device CAN be used as a BOOT DEVICE. As long as you are not planning to use the Zip drive and adapter for booting from, you don't need to do this. This process requires that the drive be located on a floppy diskette in a floppy drive, and can only be done at the time that Win XP is being installed - it can't be added in later. It uses a feature that has been a part of Windows Install routines for a long time. Early in the Install process there is a prompt to hit the F6 key if you want to install a device driver that Windows does not have already (such as a SCSI device or a RAID system), and the new SATA devices fall into this category for Win XP. If you ignore this prompt, after a wait the Install routine will proceed without it. But if you hit F6, you get to install any driver you like from a floppy disk only - you even get a chance to install more than one - before returning to the main routine. The driver you install in this way becomes one of the first things the Windows loads in its sequence at boot time, and hence allows the rest of the boot process to use that device to read and boot from.
So, with your devices hooked up as you have them now, I think (HDD as IDE Master, optical as IDE Slave, and Zip as Master on the adapter connected to a SATA port), I suggest you check the BIOS as outlined above to be sure the Zip is detected there, then let Win XP boot. Go though Device Manager to force it to Scan for a new device (the SATA port), then load the driver for it, back out and reboot. That should get the Zip drive working unless there is something odd about it. You should NOT need to re-install Win XP and use the floppy disk and F6 route (unless you want to use the Zip unit as a boot device).
If that does not work, I have one other idea, but I really doubt this is involved. Some early SATA port controllers (and it don't think Intel was among them) could not negotiate properly with later SATA II (more properly SATA 3.0 Gb/s) drives at start-up, and they could not "talk" to each other. Drive makers provided a cure for this by having a jumper setting on SATA II HDD units that forced them to fall back to the slower SATA 1.5 Gb/s communication rate. There is no such jumper setting on your IDE Zip unit. BUT maybe there is such an adjustment possible for the adapter you are using. You'd have to contact the adapter maker for this info. But as I said, I really doubt you need this.