The analyses of a text #1

The narrator recalls his childhood; he is now in custody by his stepfather in his room.
He had been living with his mother, Peggotty the only maid of his house, his very stern stepfather Mr. Murdstone, and Mr. Mudstone's eccentric elder sister in his late father's house when he was forced to go Salem House, the boarding school near London.
Now he came home after the stay during a semester.
The carrier of the cart which he have used asked him to deliver his propose to Peggotty.

...............................
But my mother made no answer, except to thank her, and Peggotty went running on in her own fashion.
"Me leave you? I think I see myself. Peggotty go away from you? I should like to catch her at it! No, no, no," said Peggotty, shaking her head, and folding her arms, "not she, my dear. It isn't that there ain't Cats would be well enough pleased if she did, but they shan't be pleased. They shall be aggravated. I'll stay with you till I am a cross cranky old woman. And when I'm too deaf, and too lame, and too blind, and to mumbly for want of teeth, to be of any use at all, even to be found fault with, then I shall go to my Davy, and ask him to take me in."
"And Peggotty," said I, "I shall be glad to see you, and I'll make you as welcome as a queen."
[David Copperfield by Charles Dickens]
I think "Cats" refers to the Mudstone.

I'd like to know what "catch her at it" here.
And I'd like to know if the underlined two "she"s means Peggotty herself.
Thank you in advance for your help.