If the cable two the internal C: drive is disconnected.
Will Vista HP recognize an external drive as C: drive?

Not if Vista is installed on that drive you disconnected. And likely not boot at all because your Master Boot Records are on your C drive.
And no, your computer will not likely try booting from your external drive unless you have your Bios set to do so.

I think my C: drive is bad. My idea is to kind of start over with a new configuration. I do not, right now, have another internal drive, but I do have an external drive. I thought to disconnect the internal drive, connect the external drive, change the BIOS to boot from first the CD drive and 2nd from the external drive. Then, boot from the CD, format the external drive and install Vista like new.

Since I have been experiencing index problems observed from CHKDSK during reboots, the new configuration with an external C: drive will either eliminate the problem and therefore prove the internal drive as bad or prove Vista bad. If the new configuration continues to display indexes problems, since the odds of both the internal and the external drives (being of different manufacturers) being bad, is higher than I can calculate.

So, what is the verdict? Should it work?

Is it possible to remove the external drive from it's caddy and use it internally??

The external drive has a circular data connector whereas the internal drive has a ribbon connector. Besides, the other end of the external drive's data cable is USB. I don't think the mobo have that kind of port on it.

Nice idea though. I don't think they make internal/external drives anymore.

Is it a purely external drive? My external is just a normal drive in a USB caddy.