
Greetings and welcome to the forum. Do please make sure to get rid of all those fears before attempting AP.
Your grandmother is privileged to have had this experience, Charles.
Good luck, Charles! Wanting to succeed is a great help. BTW don't worry about too much about vibrations. Different people experience different things: vibrations, tingling on the body, buzzing, whooshing, strange noises, etc. But when you project - you will know it!
Welcome to the forum! I think you need to deal with your stress issues first. To astrally project, you need to be both relaxed and in a state of inner harmony. The link Praveen gave should help. Best wishes to you!
Cats are excellent astral projectors! No doubt your cat was aware of what was happening to you and itself jumped astrally off your bed. I use the term "astral" in its broadest sense, as what happened sounds more like an OBE within the RTZ (Real Time Zone).
The astral body can be changed at will, so perhaps your cat had some reason to change its colour; what that reason might have been, it's difficult to say. You will have to ask your cat that telepathically!
When you returned rapidly to the physical, the cat did the same. Hence its colour was as it always has been in the physical: black!
Hope that the above helps.