Can my pets transmit infections that might be harmful to me and the baby during pregnancy?

Yes. Toxoplasmosis, most commonly acquired from cat litter boxes is the most common/concerning. You should let other members of your household handle this job throughout your pregnancy.
Yes. Complicated question. Avoid raw fish/poorly cooked meat during pregnancy. Cats crave/eat raw meat when not supervised as oppose to dogs. Toxoplasmosis among other bacteria can be found in raw meat/animals. Although harmless in healthy adults, it can cause great harm to the baby. These bacteria can be passed through the cat's droppings. Therefore, don't change the liter box when pregnant.