How should I go about introducing a new cat

[Sydneyluck]

I moved into my boyfriends home in December. A day later my cat gave birth to her kittens (cats haven’t even met yet). Now the kittens are about 9 1/2 weeks old and going to their new home (cat had 2 kittens and one of them we are keeping, other one is going to our next door neighbour).
my boyfriend has 2 adult female cats already,
One of the original cats is super shy but completely fine around my cat, The other is a huge “female dog” about it lol. Love her to death but she’s growling at the door DAILY and has gotten into a fight with my cat once already (my cat got out when the kittens were a week old so I don’t really blame them) At what age should I introduce them at for mom to be a bit calmer? Should I wait till mom and baby are spayed? and should I introduce the cats one at a time?
my cat also used to be a stray and has quite a bit of cat vs cat combat experience but she seems to be good around animals in general (however she was rehomed to me bc She scared the heck out of the previous owners 150lbs mastiff)

 

[di and bob]

There are a LOT of ideas, etc. on this site about introductions, you might research them. The main one is to give all cats blankets to lay on and switch, to exchange scents. The next step is to put up some kind of barrier, like a screen door, deer netting in a frame, etc. to let them see each other but not be able to interact. Feed them as close together on each side of the barrier as you can so they get comfortable around each other. I have had luck with harnessing the aggressor and keeping them on a leash for a while so you can pull them back, or putting like mama and kitten in a large dog kennel, with a turned over box with a door cut in it, to give them a place to hide and not feel trapped, and exchange them once in a while with the two other cats. watch carefully so that they don't turn on each other, that can happen when they are scared.Keep a stiff piece of cardboard around to stick between them if that happens. Females are very aggressive with not wanting to share their domain, and will often ignore each other when they do. This WILL happen it will just take a long time, mine took a year to get comfortable, but eventually, they did. Remember too, growling, swatting, and hissing are all perfectly normal. even a little hair flying. but watch for deep bites and scratches, THAT is not. when they start interacting, if one hides and does not interact or be around, then it is getting really hurt. if they run and hide but then come back out after a short time, that is normal.

 

[rubysmama]

Not sure if when to introduce a mom and kitten to other cats has been discussed very often on the forums, but if mom is getting spayed soon, maybe wait till that's done, so you don't risk having non-recognition aggression when she comes back from the vet.

How To Deal With Non-recognition Aggression In Cats | TheCatSite

At only 9 weeks, it should go fairly quickly introducing the kitten to the other cats. But there, again, will be the chance of non-recognition aggression, after her spay. When is that happening? Soon, or closer to 6 months?

Here's the main 2 TCS articles on introductions, which should have some ideas for you.

How To Successfully Introduce Cats: The Ultimate Guide | TheCatSite
How To Introduce A Kitten To An Older Cat | TheCatSite