The software supplied with digital cameras basically serves two distinct functions. The first is to enable the transfer of images from the camera to the PC. The second is to allow you to manipulate the images (e.g. to improve brightness, contrast, colour balance etc).
The latter functions will work with all types of images, from all sources. (i.e. as far as image manipulation is concerned you can use either of the supplied programs or, probably better anyway, something completely different).
As far as the downloading programs are concerned, they tend to be specifically designed to work with individual makes of cameras, so you might not be able to get one of these programs to work with the 'wrong' camera. (I doubt that you'd actually damage a camera; you just wouldn't be able to transfer the images).
The downloading programs are usually a pain in the @rse anyway. You usually have to have the camera turned on (which either means draining the battery or using a mains adapter). Additionally, transfer can be very slow. Hardly anyone actually uses them. A far better alternative is to buy a USB card reader. These used to be rather expensive but they're now easily available, quite cheaply, for a few pounds. (Try Asda or Tesco Extra). Using one of these card readers means that the media card, which has your pictures on, is just seen as an extra drive and you can 'drag & drop' (or 'copy & paste') images just as you would any other files.
So my advice is to ignore all of the software which came with your cameras. Buy a USB card reader for data transfer and then use something like IrfanView or Picasa for image manipulation (They're both free):
http://www.irfanview.com/
http://picasa.google.com/
Chris