So here's the story, my two year old cat Stella, got pregnant and had her kittens 6 days ago. She had them under my bed of course

. Anyways, I haven't had a cat have kittens since I was like eight, and I hardly remember it. So how early can you handle kittens? Is there some sort of grace period you give the mom or anything? I googled it, and there are many conflicting answers, some say as early as the second day, some say wait until they open their eyes.
I also feel rather uncomfortable having them under my bed, they are getting a bit more active, and I always have a paranoid feeling i'm going to step on one of them, which would be horrible, so can I move them or would this upset both the mom and the kittens?
I'm also wondering if I should get this cat spayed, when she went into heat it was all I could do to keep her inside, and she got out anyways. The problem is she has a heart defect, and can't deal with too much stress or she could have a kitty heart attack. As such my vet told me it was up to me to get her spayed or not, as the operation could kill her, but so could having babies. Luckily she was fine with this pregnancy, but I'm wondering if I should just bite the bullet and have her spayed, I hate putting more cats out into the world when there are already so many without homes.