cats need the companionship of their own kind
I don't believe so, I think it benefits some and makes things worse for others, so it depends on the cat, but for the most part unless the cat has lived with others for their entire life and is very social and attached to other cats I don't think they NEED them.
.when another cat enters the picture,the one who's lived in that home will 'show' the newcomer 'the ropes'
I'm not entirely sure what you mean here, could you explain further?
.a fixed male,and a fixed female get along great
If two cats are going to get along this would be the best match, but putting these two together does not mean they will like each other, some cats don't like others. I believe I heard the hardest cats to put together are two females or two intact males. Neutered males are supposed to be pretty easy to match. Personality is important as well, a lazy cat would not be good with an overly demanding kitten as he would drive him insane. Plus an active cat would benefit from a cat to play with, not one who does not want to play so I think personality is more important than gender.
.cats MUST fight,it's how they determine kittyarchy-and when a new cat comes along it upsets the order,so they have to fight again
No they don't have to fight, mine have never had a legit fight, they have gotten angry and bitten pretty hard (no blood though) but have never had a spitting, hissing, swatting fight. Cats (and dogs for that matter) can use a lot of other methods, without ever touching the other cat, to show who is in charge, body language is very important to cats.
You'll have to decide for yourself, how long was each of the cats with you for (and how old was your cat at the time)? Think about each cats gender, age, size (consider hair length as well, fluffy cats look larger), personality etc. and how your cat felt about them, one of those factors may have contributed to how he felt about them. I think if you want another cat then give it a try, perhaps a little female would be best for him. You could try fostering first to get a better idea of what he likes, perhaps you'll even end up fostering the kitty you keep. Remember to do a proper intro, not just toss them together, and for a while keep the newbie out of your cats "special places" like his bed. Keep them separate at night and when you are not home until they like each other.
Good luck, I hope you get a new cat!! Any petfinder kitties you have your eye on?