Could an early delivery (week 37) affect my baby's health in the future?

Not usually. Most of the issues involved with a 37 wk delivery will show up in the first 24hrs. After the initial period of stabilization has passed , they are essentially like any term newborn.
Question. I hate the term, but did you and your OB discuss your " advanced maternal age, " and dad's possible "advanced paternal age"? I've taken care of children whose parents thought that prenatal chromosome testing and fetal ultrasounds " told you everything." no need to fret, age is a risk factor, not a given. Just pick a pediatrician who does developmental surveillance with standardized screens.
Rarely. While babies today do very well at 37 weeks, it does not mean that there cannot be some rare occurences of some babies with infections or breathing problems because they are delivered early.