In my personal experience, having that many animals is challenging but the burden is lessened if the cats like each other. If you can throw some money at the issue (which I understand nobody can) those litter robot things are a miracle for consolidating how many boxes you need and how often you have to tend to them. Room fans and HEPA filters can cut down on smell and the fur problem. Aside from that, it sounds like playing caretaker is probably the primary stress inducer, you just can't do anything immediate about that obviously. Make sure you and your partner's communication is in a good place and just know you'll adapt eventually. How much space you working with?

automatic feeders maybe? i have four cats(no dogs) and all are on the same diet luckily. doing their auto feeders, water bowls(4 water bowls) and litter boxes(i have 5 boxes) takes me maybe 15 minutes total. the feeders turn 3 times a day and it has 4 slots. I have the Orsda brand from Amazon. i clean the litter boxes once a day. I buy dog poop bags and put the litter in there and then it goes into a special can outside and i put that bag in the normal trash on trash day.

it's more expensive but two of my boxes are using crystal litter(Pretty Litter) because one cat has been using it for a long time, so i just scoop the poop out of those and then you have to stir it around for the urine. Pretty litter gets changed out every 30 days.

Cats need to be slowly and carefully introduced. If you just throw them into the same space right away they'll often fight and may hate each other indefinitely. On the other hand, if you keep them in neighboring rooms, they can get used to each other's smells and play with each other under the door for a week or three.

You'll have an idea for how they're feeling if you pet the cats in one room a lot and then let the cats in the other room smell your hand. If they're reactive in a negative way, they're definitely not ready.

After they seem fairly comfortable with their smells and playing under the door, you can transition to something like a screen door (cats will just jump over one of those baby wall things) or just short glimpses through the open door, and just ease into it that way. Maybe 1 on 1 cat play dates. I can't emphasize enough that cats flip out with too sudden of introductions and in scenarios where they feel threatened, like being outnumbered. Also being in a new space makes them on edge.

Anyways the point of this recommendation is that it is infinitely easier to take care of 4 cats that get along with each other after a 0.5-2 month acclimation than having to separate them indefinitely because they hate each other.

Not sure what to do about the dog. Dogs can be friendly to cats they were raised with and acclimated to but be really into chasing introduced cats. I haven't had to introduce adult cats to adult dogs before, so do a lot of reading!

It isn't really that many animals, if no one wants to budge then do the obvious and get rid of the mother in law.

Vacuum robot helps a bit, too, less dealing with animal fur. Other than that, I second the automatic feeder and automatic water bowl. Maybe an automatic litter box, too, though I have no experience with those