Question

Would the COX1 gene work for DNA barcoding of prokaryotic organisms? Why or why not?

Answer 1

I think, it might be true, because in biology we are dealing with alive organisms which are the results of evolution, not computers!.This is one of the the most widely used genes for that reason and also simply because early studies.
No gene is species specific Evolutionary processes proceed without "knowledge' of human imposed species boundaries.

1. High mutation rate due to limited repair system (5-10 times that of nuclear DNA).
This is because mitochondria are essentially bacteria (prokaryotes) living within eukaryotic cells. Bacteria have less DNA damage and repair machinery than the average eukaryote, and mitochondria have even less because they have lost most of their genes. The mitochondria import a lot of proteins and metabolites that they need from the eukaryotic cytoplasm. but they don't get all the DNA repair enzymes.

Conserved structure of the genome.
It is difficult to create a multiple sequence alignment when the gene order is different between the organisms being studied. Over time, bacterial genomes tend to get scrambled with chromosomal inverstion, insertions, deletions, horizontal transfer by phages, etc. The mitochondiral genomes tend to be rather stable within a given group of related eukaryotes such as the mammals, so it is easy to align them for phylogenetic analyses.

Lack of recombination.
Bacterial DNA (and eukaryotic nuclear DNA) is full of recombination events and this confounds phylogenetic reconstruction. Humans don't recombine with horses or frogs of course, but we did rather recently receive alleles from Neanderthals after being separated from that lineage for a long time, and when we split from the chimpanzees and gorillas there was incomplete lineage sorting which means that different genes in the human genome have different histories.

High phylogenetic signals of COI (cox1) than other genes of mitochondria.
Different genes, and even different regions of a single gene, evolve at different rates. Site that evolve quickly are good for seeing the difference between individuals of a species or between subspecies, but not so good at deep relationships (comparing mammals to fish for example) because too much divergence leads to more "noise" than "signal" in the data. Highly conserved regions are better for studying the deeper relationships.