Can I make an image of my existing hard drive on a new external drive, then fix the BIOS so the external drive is the "C" drive?

In theory it is possible, but have never tried. Only thing you can try is to image the drive with what software you have and then set bios to boot to the USB -

Your BIOS must support booting from USB

Don't do anything with internal drive as above may not work

I do this on a regular basis in my business. I use Acronis True Image software to create the image because it supports almost all external USB drives (I'm sure there are others - this is just my preference because of its diversity). The biggest issue will be as stated by seahwk - will your computer boot from USB. You will have to make sure that PRE-OS USB support is available and enabled in your bios and then that the external usb drive is active as an available boot device, and then set it as the primary boot device.

Yup and yup to this :)

Yes I am having problem istalling the software
For my printer hp2575 photosmart all in one Their a problem in the c drive