For now I will only touch on the second half of your post about the gaining of information. I would not necessarily call this a gain an information more then it is simply a rearrangement of existing information. The bacteria rearranged existing information to be able to break down lactose. I'll refer you to another paper on the Evolutionary Potential of the ebgA gene (the replacement gene for the lacZ gene which was deleted).
Evolutionary potential of the ebgA gene. | Molecular Biology and Evolution | Oxford Academic
Interestingly, the study finds that if the lacZ gene and ebgA gene are deleted there is no chance of utilization of lactose. There is also a 35.5% sequence identity between the 2 genes as well as the proteins sharing 13 of 15 active site residues, giving credence, at least to me, that no new information has been provided rather there has simply been a rearrangement of existing information. Also, quite interesting is that the wild-type enzyme of ebgA is quite ineffective in utilizing lactose and its analog lactulose. So ineffective that these sugars cannot be utilized for growth when the operon is expressed constitutively at a level that the ebg protein constitutes 5% of the cells soluble protein (the ebgR- repressor gene was removed).
The evolutionary potential of the gene was then tested when the detla lacZ ebgR- is subject to selection on a plate of lactose. Only 2 phenotypic classes of single step spontaneous mutations were obtained, Class I and class II. Class I will grow on lactose but not on lactulose, and class II will grown on lactose but on only moderately on lactulose. No other class was found to utilize lactose or lactulose. The conclusion was that the ebg operon is quite limited in its evolutionary potential. So your example shows not only an "evolved gene" that in wild type has only feeble beta galactosidase activity towards lactose and lactulose, but also an evolved gene who's mutations to increase its activity towards lactose and lactulose, which constitutes a whopping number of 2, are hardly convincing as well.