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Hi. Welcome to Just Answer. My name is***** Thank you for your question. I'm going to say that the answer to your question will vary in different situations depending on the dog(s) and cat(s) involved. In my own house, I have one older dog that gets along (and always has) gotten along great with other cats. Currently in our house right now we have 4 cats that she absolutely gets along great with. Now, does she like other dogs? No. She can be pretty dog aggressive. I know of lots of houses that have dogs and cats in them where they get along very well. Does ever dog do well around cats? No. Some can be very aggressive and not do well around them.
Here is a just one good link on going over how to introduce dogs and cats together in a house.
If you are going to consider doing this / taking in this small stray cat, I would take him to a vet's office first for a check over as you don't want to bring in ear mites or fleas into the apartment. They can check him over to make sure that nothing like this is brought in.