We face each other when we take our last breath and our heart makebeats its last beat. We face each other when we take our last breath and our heart beats its last beat.

I'm not sure the verb "makes" works in this context, but I'm also not sure what I'd suggested instead...

It would be dishonest to say that we meet death only once, whereas our story comes to an end. It would be dishonest to say that we meet death only once, as our story comes to an end.

"Where" isn't wrong, strictly speaking, but this sounds more natural

If every memory of me disappeared tomorrow, I think only a minority of people would notice. If every memory of me disappeared tomorrow, I think only a minority of people would notice.

"Minority" really only makes sense when there's a larger reference group in question.

My life isn’t full of people,; I don’t have a lot of friends or acquaintances who would keep memories of me alive for a long time. My life isn’t full of people; I don’t have a lot of friends or acquaintances who would keep memories of me alive for a long time.

In my life I haven’t done anything particularly brave or incredible, besides,and I also I haven’t won anything, so that people would think about ithat either. In my life I haven’t done anything particularly brave or incredible, and I also I haven’t won anything, so people would think about that either.

Besides is for a different point, but I think this point flows on well so I'd just use "and"

In my life I haven’t done anything particularly brave or incredible, b in my life. Besides, I haven’t won anything so that people would think about it. I haven’t done anything particularly brave or incredible in my life. Besides, I haven’t won anything that people would think about.

The second sentence sounds somewhat unnatural, but I'm not sure quite what you mean, so it's hard to give a specific suggestion.

Moreover, for every person I wasI expressinged myself differently to every person I met. Moreover, I expressed myself differently to every person I met.

For some people, I was a sister, a daughter, a friend, a student, or a competitor; but for some reason I’ve never been myself. For some people, I was a sister, a daughter, a friend, a student, or a competitor; but for some reason I’ve never been myself.

a comma after "competitor" would also work, but I think the semicolon is nice for readability

However, the truth is that life ends when memories of us are no longer remembered. However, the truth is that life ends when memories of us are no longer remembered.

This sounds a little redundant. You could perhaps say "... when no memories of us remain" or "... when no one remembers us any longer."

I have realised that memory isn’t supposed to be about something material, but about feelings. I have realised that memory isn’t supposed to be about something material, but about feelings.

I think the closest people would remember my kindness, love, and passion. I think the closest people would remember my kindness, love, and passion.

But also - the dream. But also - the dream.

I'm a little unclear what you mean in this sentence. They would remember your dreams? They would remember your passions? They would dream of you? If it is the first one, I'd phrase it as: "But also - my dreams."

Before death comes for me, I want to do as much as possible to be remembered, respected, and loved. Before death comes for me, I want to do as much as possible to be remembered, respected, and loved.