Do people with Dementia realize they are this way? Or is their life as "they are living it" seem pretty normal?

Each individual is a bit different and much depends on the type of dementia they have as well as the stage the are in.
However, in most cases when the people are in a memory unit they have passed into a stage where their own reality is the only one they know.
Again, depending on the stage they are in, they may become very frustrated when they forget something they want to remember. It's also common for them to be frightened because paranoia is often part of the disease. For these reasons they may not always be content.
However they may have times of serenity if they are getting proper, individual care. Especially in the later stages of Alzheimer's, you'll see many people totally in their own world. However, I'd never want to take the chance that they don't know what's happening around them. That's why it's important to not speak negatively when they are within hearing distance.
Treating them with dignity and respect is important, and you sound like a person who will do that. It's wonderful of you to care so much.
Take care,
Carol

Carol's answer is on the money from what I have observed. I witness many different types of Dementia. The one thing I do see, is that expressions, body language and emotions of others is a trigger for a person suffering from dementia. So if you act as if something is upsetting you a person with dementia gets upset. If you act happy and calm they remain happy and calm. I see a woman holding a baby doll happy as can be, but when her daughter visits she tries to get her Mom focused on her visiting and not the doll. Then her Mom gets upset because her daughter tries to bring her Mom to her own reality instead of accepting her Mom's reality.

My husband lived at home with Lewy Body Dementia for 9.5 years. He was never in a memory unit or any kind of a care center. For the most part he was quite aware that he had dementia and that it was the cause of his limitations. We personalized the disease and referred to what Lewy was up to. "Hon, you have a very good brain and you are a very smart man, but Lewy seems to be up to his tricks today. Don't worry about forgetting things today. I'll be your memory ... and maybe tomorrow you can be mine!"

In some ways his awareness of his disease made life easier for him and certainly for me. In other ways that is a bitter knowledge. My sister said, "If I ever go gaga I hope I just get there and I don't have to go through a journey where I know about it."

If something had happened to me and my husband had to go into a memory care place he would know why he was there. I think that Carol is right that it depends on the particular kind of dementia and the stage they are at. You probably can't generalize about the people you see. Getting to know them as individuals will answer your questions.