I don't know shit about brands BUT I've had a Serta (I think) ever since I can remember, and it's done alright. I've got a queen size on top of boxsprings, and I've always had it all just laying on the floor, no frame or anything. The other day I noticed it's starting to sag a bit in the middle, but I can't complain considering I've had the bed for probably a decade. The big thing for me is getting the right pillows, it can make a huge difference in how well you sleep due to your spine angle.
My queen size bed barely fits me, and you are a couple inches taller than me. When you go shopping, I recommend asking about longer queen size, or even a king size to accomadate for your height. I've heard of being able to get odd size mattresses, but I really don't know that much about it.