Without an ozone layer this cell would receive heavy doses radiation from the sun- obviously this would kill it.
You answered this in the first part of your paragraph - the ocean protects it, if it's deep enough.
Somehow this cell also had a mutation that caused it to be able to reproduce asexually- if it didn't have this mutation- we wouldn't be here right now.
Since asexual reprodction would have to be the first thing it could do - replication being a prerequisite of being considered "alive" - it doesn't make sense to say this comes from a "mutation." Rather, asexual reproduction is an inherent quality of life. If it can't reproduce, it's not alive.
If we're talking about the first living thing, reproduction is already a given.
Lets say that chemicals in this primitive ocean make millions and millions of cells complete with all organelles- NOT from asexual reproduction but because of some kinda chemical reaction.
All life processes, including reproduction of any kind, are "some kind of chemical process" so this statement is more or less meaningless.
Given that there are plenty of living things today that lack all organelles why would you propose that the first living thing would have to have all organelles?