Very short perineum. Family history of big babies (>9lbs). Don't want to risk serious tear and damage. Can I ask for c-section? Terrified of natural!

You can ask. Prelabor cesarean section by maternal request accounts for at least 5% of all cesarean sections in the U.S. At this time. Many obstetricians object to this, but most will be fine with such a request and honor it after appropriate counseling, especially if you are only planning to have 1-2 children. Talk to your obstetrician - and good luck!