Their claims are based around an experiment in which they either have an empty expression vector or a random sequence of 150bp inserted into the vector. The 25% beneficial claim comes from those that perform better than the empty vector but the empty vector is not a true negative control; rather it still contains the FLAG tag so rather than comparing whether random sequences are beneficial per se, it is really comparing whether they are better or worse than having a different sequence before the FLAG tag is better or worse than having the FLAG tag alone. Also, I can see no evidence that they controlled for plasmid copy number, and I think the plasmid they're using can have variable copy number.
I find it particularly telling that only a single sequence performed less well when a stop codon was inserted at the start of the sequence.
I would think that an increase in fitness between a FLAG tag and a FLAG fusion protein (or RNA molecule) would indicate that the added 150 base pairs is responsible for the increase in fitness.
It does indicate an increase in fitness, but it does not indicate that the random DNA sequence is coding for anything of consequence. They want to argue that the sequences they've introduced are doing something of consequence. I think the more obvious interpretation is that they are not doing anything in, and of, themselves but rather preventing a harmful effect.
As to copy number, it would appear to be a pretty standard expression vector which would have multiple copies and very high induced expression. My criticism is that the introduced gene may make up a disproportionate percentage of total RNA or total protein.
Quite. But they measure (organism) fitness using the proxy of the number of copies of the plasmid in their population; if the plasmid varies systematically in copy number that will bias their results.
As to stop codons, the gene could still have activity as an RNA gene, so I don't see too much trouble with the translated peptide not having activity.
RNA can certainly have activity; however the majority of active sequences in the body are protein coding. If you're claiming that you have hundreds of beneficial sequences but only one is influenced by the STOP codon that implies a ratio of active protein sequences to RNA sequences I find implausible.