I don't understand cats

Dogs, you know, are such an easy study.  If they do that little stretch-bowing thing, they want out, or they want to go for a walk.  If they want to play, they bring you sticks and balls.  Pet them and they are ecstatic.  There is a reason for everything a dog does, right down to rolling in carrion.  
But cats?  There is no one-fits-all description.
When I was a little girl I had several cats at different times.  One even slept with me, and woke me up if she wanted to go out in the middle of the night.  I think she's the one that had kittens in a basket of clean laundry.  Anyhow, back then all cats seemed friendly and normal.  None of them rejected my affection.
Either I was missing something, or I have changed.  Or cats have.
I have three neutered (and spayed) barn cats.

There's Jake, the male.  He has gotten so huge he can barely get through the openings in the milking barn where the cats come and go.  He seems friendly enough until you reach down to pet him; then he rolls on his back and exposes his belly.  If you start to rub his belly, he attacks your hand with teeth and claws.  And do NOT stroke his back while he is eating his breakfast.  I have no idea why he has this attitude.

There's Suzy, who is so wily I can't even get a closeup picture of her, although I will admit things happened to her in the past that would make anybody suspicious.  Don't ask.  Anyway, if I have lots of time to spare, I can sit on a hay-bale in the barn and say "kitty kitty" over and over for twenty minutes, she might deign to come over and rub against my leg; if I move my hand toward her very slowly, she might allow me to stroke her.

My friendliest cat is the mother of the above-mentioned felines.  If I had immediately taken her in after she was abandoned by former neighbors, she would have been a cuddly, friendly cat.  However, I didn't want a mangy cat around climbing my screens and spraying my porch, so I ran her off.  She and her scrawny, stunted kittens lived on field mice and whatever she could scrounge out of our trash barrel.  Her life was not an easy one.
Once I realized what a huntress she was, I switched tactics and started feeding her.  The sickly kittens she had at the time didn't make it, but she did.  She had once been a little girl's pet, so she would let me pet her and pick her up at times, but she was wary.  She remembered the times I had hissed at her and run her off the porch or out of the barn.

Obviously it hasn't kept her off the porch, but now that I know she isn't going to climb on my window screens and tear them up, I don't mind.  Why do cats always want to sit in my chair?  And if I leave a coat outside, why do they assume it is their own personal bed?  Cats assume too much, in my opinion.  
Mama Kitty takes a walk with us every day, which I think is rather sweet.  She goes up and down every hill, turning around when we do; she goes the whole two miles or more.  Sometimes she runs ahead of us just so she can lay down and roll in front of us.  Why?  Does it feel good to stretch and contort her body along the ground?  For that matter, why would she want to go for a walk?



Why do they do what they do?