If I was born prematurely, can I have a normal pregnancy?

Yes. Women who were born early or whose mothers had pregnancy complications or difficult births should tell their OB providers about those events. But in general, the likelihood of the daughters having those problems is less than their chance of not having them. A detailed history from women in the family is helpful for everyone pregnant whether or not their moms had high-risk OB or other conditions.
Yes. Women who were born early are rarely more likely to deliver early unless they have other risk factors. Even with a risk higher than that of the general population, their having a preterm delivery is not 100% certain. Women whose moms had major OB complications should tell their doctors. But usually the likelihood of the daughters having those problems is less than their chance of not having them.
Yes. The great majority of women born as preemies can have normal pregnancies. Exceptions would be women who were very ill as preemies and ended up with long-term medical problems. Legs or hips contractures from cerebral palsy make it harder to get into a good position for vaginal delivery. Short gut syndrome after necrotizing enterocolitis makes it harder for a pregnant woman to get enough nutrition.