Does his stress level affect my chances of getting pregnant?

Yes. Stress can cause men's testosterone to decrease and therefore can have an indirect effect on sperm counts. Stress increases the level of reactive oxygen species in the semen and this can hurt sperm. Stress also often indicates poor lifestyle choices; loss of sleep, bad nutrition and other factors that can harm sperm. Have your partner get a semen analysis.
It can. Stress can affect different people in different ways. It can affect levels of hormones such as dopamine, cortisol, epinephrine, serotonin, etc which in turn can adversely affect fertility.