Going down the stairs easily, foot over foot, does come slowly. It gradually gets better. At first (at about 3+ months) I could do it slowly and with care. Gradually, I grew faster and more confident. I still hold onto the hand rail if there is one, because I don't want to fall, but I could probably manage without it now. It's just my little safety precaution.
I can cut my own toenails, but it wasn't easy at first, because it demands both knee flexibility and the ability to bend over. I do it now by putting one foot up on the side of the bath. I think it is possible to get long-handled scissors, but I wasn't certain I could position them correctly, so I didn't buy them.
At first, I asked my husband to cut my nails. He balked at that, so I went and had a pedicure.
Your limping should decrease gradually.
It helps to walk normally if your knee has full extension. If you have access to a full-length mirror, watch yourself walking, and concentrate on doing it with a correct gait. If you still go to PT, your therapist may be able to help, too.
Your speed will increase gradually as well. Remember that your knee has been through a bad time and it needs time to recover.