>>529442442
Consult the list first, Nice Nature and Goldship gave Rudolf double circles, which gives a greater chance at inheriting skills. The rest is luck. Try to get your desired stat as high as possible, 600 for a 5% chance at a 3 and >1100 for a 10%.

>>529227427
You can get the pace chaser 1-3* gene if you have pace chaser A or S by the end of your career. This is a binary decision: if you are pace chaser B, its impossible to get the gene. In terms of odds, it is fully randomly selected among valid distance/turf/strategies.
>Nice Nature with Spe?
Fixed internal number. See image, affinity decreases moving to the right. Affinity increases the likelihood of rolling genes (skills and aptitudes) during the inheritance events in April.
FWIW affinity is later buffed so its trivially easy to get "high affinity" with everyone