The stethoscope James uses to examine the cow is a late 20th century model, and thus not one used in the 1930s.
At approximately 51:58, Siegfried rests his feet on the table. The soles of the boots he is wearing appear to have Vibram's original "Carrarmato" lug sole design, which was patented in Italy in1937 as the first boot sole with rubber lugs. Purposefully designed for mountaineering, no person outside of Italy would have had access to early prototypes or production examples of this boot in 1937, the year in which Season 1 is set.
Tristan places a bet on a horse because he knows the favorite will not make it to the race; when he wins the barmaid who is making the book happily pays him and is happy because she doesn't need to pay all her other customers who bet on the favorite. In reality, when a horse is scratched the bettors get their bets refunded--she would have no reason to be happy and every reason to be angry at Tristan for betting with inside information--essentially stealing her money.