Breast milk is inherently sterile (unless the mother has a severe bacterial infection). It usually gets contaminated with the skin microbiota during feeding.
So no, I wouldn't count on L. reuteri being found in a random sample of breast milk.
That's hotly debated. Even the study you cite cannot rule out skin contamination.
But unequivocally showing the existence of entero-mammary pathway (translocation of maternal gut bacteria from the gut to the mammary glands) would be very exciting news indeed.
Yes, because if you are buying breast milk from a woman who used a breast pump, then the skin contamination is very very little compared to actual breast suckling.
You've put more confidence in your statement than the scientific consensus allows.
A brain microbiome is hotly debated, no concrete proof (yet?). I personally also have a hard time believing in a healthy bacterial blood microbiota, but it's also proposed.
So no, I wouldn't count on L. reuteri being found in a random sample of breast milk.