Answering the rest of your post is a bit more complicated.
Would a single cell that reproduces asexually and creates a community of other like organisms that are exposed to the same environmental conditions ended up having the same mutations across the population.
What we tend to see if we culture a starting strain of bacteria and run it through multiple generations is the emergence of multiple strains due to small differences in a number of genes. In this experiment one gene was put under heavy selection pressure. Higher variation tends to emerge in genes that are not under great selective pressure. Genes with high selection pressure are more likely to have highly conserved sequences across species and perhaps even across kingdoms. For instance, our ribosomes have regions that are conserved fairly well across eukaryotes, bacteria, and archae. But it's impossible to put every gene under high selective pressure, so a variety of alleles (possible sequences for a gene) will eventually arise. This is one way we can estimate how old a population is--if it has low variation in alleles, it probably arose from a bottleneck event not too long ago. On the other hand, Africa is thought to be the region that humans first appeared in partly because Africans have a much higher genetic variation than people from other regions, indicating an old population. So my prediction is that if they repeated this experiment they would again have the same mutations appear, and one (possibly not the same one as last time because there's a certain margin of chance if the top two variants aren't very different in effectiveness) would become prevalent. If they then continued the experiment they would see this strain diverging into multiple strains that have the same allele for the gene required for life at high temperatures, but they would have different alleles for other genes. Unless one of these happened to have some other advantage (better at segregating rare metal ions, for instance) disappearance of various alleles would be driven by random genetic drift and new ones due to mutation would replace them.
Darwin tended to emphasize competition, but sometimes lack of competition can kickstart the process of evolution. Some of the most dramatic radiations we see occurred after mass extinctions emptied a variety of niches and allowed replacement by new species.
Interesting! Thank you.