There is a first time for everything, I've not come across a Bios before with no mode options for the hard drive, most have several to choose from including various RAID options.
I have just looked here: http://h10025.www1.hp.com/ewfrf/wc/softwareCategory?os=228&lc=en&cc=us&dlc=en&sw_lang=&product=3184140#N402
There are no SATA drivers available for your PC, so HP obviously don't make provision for anyone who wants to run this model of PC using a Retail version of XP. The SATA drivers will be on Recovery discs which they do provide, but you have to pay for them. You will see on the above link that you can order the discs if you want them.
The only thing I can suggest is to follow the link I gave earlier to integrate the SATA drivers onto a copy of your XP disc. In that link there is a very comprehensive download with a big collection of SATA drivers, hopefully one of them will be compatible with your motherboard, but you won't know for sure unless you try it.
When you have run a Repair install the hard drive would still have the SATA drivers installed allowing the process to take place. When you run a clean install the system can only load drivers from the XP disc so without the SATA drivers it cannot connect with the hard drive. As there is clearly something wrong on the hard drive this is also causing a problem when trying to run the Repair Install.
From your post No5.
HP is a scumbag company that was too cheap to send out recovery/windows disks for their xp computers. They only have a destructive windows repair which is essentially a format + reinstall.
This is the case with the vast majority (possibly all) OEM machines not just HP. When you first started the machine from new there would have been a pop up to ask if you wish to make a set of Recovery Discs, either that or it would have been explained in the user manual as one of the jobs you should do on your first boot. I could be wrong here but that is usually the case.
Whatever you do I would advise you to run the hard drive diagnostics which I gave instructions for as this problem may all have been caused by a failing hard drive and even if you get the SATA drivers integrated onto a copy of XP the process will not succeed if the drive is faulty.
Installing Windows 7 would not have this problem as it includes SATA support so you could upgrade without a problem.