To answer your first question, it is possible, but inadvisable to try to test this power supply. It is a proprietary wiring setup HP has engineered. The PSU provides only 12v to the motherboard, where it then converts the power and all devices (hard drives, cd drives, fans, etc) connect to the motherboard directly. Simply put, part of what is in a normal PSU is actually in the motherboard itself. It looks like they even changed how the power on signal is sent from/to the motherboard, again, it might cause the power supply and/or you harm trying to even test this.
To answer your second question, it is VERY unlikely you would be able to fit any standard power supply in that case without some rigging, and even then, you would not be able to connect it to the motherboard without some rigging and adaptors. Also, since this power supply is often used in HP slim desktops, your case might not even be big enough for a standard power supply. Just buy an identical power supply (about $50US) and call it done.
Here is another discussion or replacing the power supply: What is the 6-pin power connector on an HP ProDesk?