Maybe you were more relaxed because you didn't see her? And Daisy felt that and so she was relaxed too? It sounds like you are making progress. I think with time things will get better.
I know for me, I prefer to just put my guys in a sit stay until the person passes. They usually respect that better than if I were to rush past. If I keep moving, then they keep moving and sometimes that means pulling/lunging. For a dog like Daisy (and quite frankly, Burton still) you can grab some treats and work on 'attention'. Ask for their name, get eye contact and tell them how great they are the whole time. Treat them or hold the treat. If you are feeling fancy you can use some human food like cheese and stash a piece in your mouth. And if you are feeling more fancy than that, you can launch the cheese from your mouth to hers. Sounds so weird but it works so well to keep eye contact. You'll notice they will look at your face rather than your hands. It keeps your dog working and keeps her attention on you rather than her surroundings.
Let me know if you try it and how it works. You can even start small in places you know aren't as frustrating. Maybe just outside somewhere, maybe even just in your house (use a ball or favorite toy as a distraction).
Oh also, with big chunks, I remember when Charlie was in puppy soc, we used bigger chunks of freeze dried liver to keep his attention. Say he would be in a sit-stay, we would give him 'tastes' of the liver, let him nibble and pull the treat away. Then he knew I had it in my hand and wanted more.
Good luck!
With the cats, I would start slow. Gate some areas of the house so the cats have safe places, give them towers or places to go up high. You might be more comfortable at first keeping Daisy on leash so you can correct her or limit her space. Tell her how great she is when she ignores the cat or allows the cat to move freely without chasing her.