They tested their hypothesis by BASICALLY placing a mutant neurospora (after being spammed with x-rays) into multiple mediums each with one amino acid missing, and found that it only grew in one (tyrosine or arginine etc). This validated their hypothesis as it showed that one gene is responsible for one protein, but they had to change it to one gene one polypeptide because proteins can be made of multiple polypeptides hence multiple genes.
Pedigree analysis is useful for advising families about chances of disease, blood types or hereditary traits of future offspring. For example one question I've seen asked what is the chance of the child getting a disease, which can be used to inform families etc. Also it can be used to deduce the type of trait it is, e.g. sex-linked, codominant, and also if it is dominant or recessive etc.