It is so exhausting! Those of us who have had puppies have been there and understand. It will pass! With my Luna, I used a puppy pen in the garden for toileting because she would get distracted otherwise. It worked great; I'd pop her in, she'd go and, during the day she got a little play in the garden as a reward, at night straight back into the house.
Even the cat chasing will just be a case of gaining a little maturity, I'm sure. It's tough right now, but she's a tiny baby with very little capability to regulate herself. I didn't have to contend with a cat, but Squidge was a really bad leaf chaser on walks. They were SO EXCITING! Now, of course, they hold no interest to her whatsoever. It'll just be an age thing, and one day you'll look back and realise it's not happened in a while.