I have concerns about dying daily, and perhaps hourly. I don't consider myself a depressed person, but someone who is at a philosophical crossroads. I can't come to terms with the finite nature of our existence and I am not sure I would characterize someone who was ok with it as "normal". I would say complacent, perhaps unimaginative, or too focused on their day to day affairs. I would love to rid myself of this rumination, but it's been with me all my life.
You are describing exactly me before my depression treatment kicked in. I would also call that a "rumination" and there were times when I though being one of the few people who SEE what the world really is like was somehow good for me.
It wasn't. In my case it was depression/axiety/OCD. I still know that life is a paradox, but I don't ruminate about it since there isn't anything I can change by thinking about it all the time.
There is a great quote from Ernest Becker:
"The neurotic opts out of life because he is having trouble maintaining his illusions about it, which proves nothing less than that life is possible only with illusions."