Yes! I have had it happen several times. I love dreaming and I even feel emotions and makes decisions in my dreams. Sometimes it can feel like an out of body experience. You are not alone!
When I dream a lot, I find that I wake up extra tired and feel crappy the next day, usually with a headache. So I think it’s normal.
Usually not, but it does depend on what you dreamt of, and how much you (tried to) control and manipulate the dream. Lucidity itself doesn’t make you tired. Just think of how tiring nightmares are!
It could be your new mattress, if you think it can be a cause. Also, if you sleep for too long, you can actually end up quite tired!