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Even if possible the question would be if it made sense to live forever and in addition give birth to new humans. There is already a huge overpopulation on earth. Where would that lead?
And I only wrote about the biological/genetic aspects of the topic. The ethical are complicated too: for example there is the scenario that technological progress would make immortality possible but only very rich people could afford the necessary "treatment" ...

Overpopualtion is overstated problem in my opinion. Longer functional lifespans will allow humanity to spread out across the universe. There are always going to be big ethical issues.

In my opinion overpopulation is not overstated - but I appreciate the fact that different people who have the same information come to different conclusions. That's what makes discussions interesting. :)
I come to my conclusion when I think about how fast we exploit the limited resources available (the amount of clean water, the number of edible fishes in the oceans, the rain forests - nearly all important resources are decreasing). Yes, we may find new ways ... start to eat insects, farmed algae or artificial food from cell cultures for example, but finally the resources are limited and we are getting more and more.
Your idea to conquer the universe is really fascinating, but considering the enormous distances I have some doubts if it will come true (and if it will be done successfully). We (or better to say our descendants) will see.
By the way, if I was god (not that I am sure that there does exist one) or whoever who constructed the universe for whatever purpose I also would have made we distances as huge as they are to make sure that such an aggressive species like Homo sapiens stays restricted to a very limited area. :-)