You will be fine without the brace as long as you don't do anything stupid or have something extremely unusual happen. The exercise you do that hurts, is it like a leg extension or are you lifting a straight leg while seated more like a leg lift? I had kneecap pain that I described as feeling like it wasn't "tracking" right. When I'd bend my leg as far as I could, when I'd work to straighten it out, my kneecap felt like it had a catch until I got past a certain point. I think I've figured it out in that my quad wasn't strong enough and my patella tendon was pulling it in a way the leg couldn't compensate. Now that my quads are getting stronger, it doesn't hurt at all. Not sure if that's what you are going through with your kneecap.
I am doing pretty well with stairs actually. It was obviously easier to manage the stairs if I led with the strong leg on the way up and the repaired leg on the way down but if you are talking about alternating like a "normal" person, I think it has both to do with leg strength and ROM. Practice will definitely help but I think the exercise that helped me most is the one where I would sit on a stool or in a chair with rollers on the bottom and would pull myself about 8 feet using only the heel of my injured leg and then when I'd get to the end, I'd have to use the injured leg to push back to the starting point. The PT would have me do that for 5 minutes and it seemed to help with both strength and ROM. Most people doing it that have knee injuries were doing it with both legs and going ridiculously slow. I'd try to do it as many times as I could and was sweating like a mofo. My best was to go up and back 38 times in 5 minutes and that's when the PT had me go to single leg. He doesn't even have me doing that exercise at all any more and I'm doing step up and step downs which will ultimately help the most and I wouldn't recommend until you have some strength starting to build up because they are both very demanding exercises. These started with holding the handrail on the leveled out treadmill and using only the bad leg to step up. The trick is to have the good leg as close to the step as you can and lift your toe off of the ground on a straight good leg. This makes your repaired quad do all of the lifting and it's pretty brutal. I try to explode up (more like a weak firecracker) and gradually step down and it's more like a controlled fall. The step downs are crazy difficult. While standing on the treadmill's edge, I lift the toe of the good leg and try to lower that leg so that my heel touches the ground and immediately go back up. My leg shakes like a jackhammer. My PT told me not to do anything that hurts and I said that it's just weak but didn't hurt at all. He said it was hurting him to watch me and he walked off while I kept doing them. Both of these exercises have continued to make stairs more easy, but again, I'd ask the PT about them before doing them.