HOW CAN I INTRODUCE A NEW DOG SAFELY WHEN MY OLD DOG DIED RECENTLY WHO GOT WELL WITH MY CAT?

The best solution it would be better to introduce a pup into the household than an adult dog. This is because a puppy is likely to be more accepting of existing pets and regard them as being higher up in the pack hierarchy than him. Your cat will also view a puppy as less of threat than an adult dog and be more relaxed about the newcomer.