Can a woman who is very hairy have problems giving birth?

No problem. It depends on the cause of the hair growth but if it's genetic (you're always hairier than others) and not due to a hormone producing tumor, then no problem. If you are already pregnant, then you don't have the tumor mentioned above, so again, no problem giving birth.
NO. Amount of hair anywhere on the body will have not effect on birthing a baby. This may require some shaving to promote a better effect (rare). Interesting question but categorically no.
Unlikely. Any really bad disorder (like testosterone producing cancers) that can cause excess hair is likely to preclude pregnancy...Milder disorders like pcos makes it difficult to get pregnant and u'd need to pay attn to gestational diabetes and excessive weight gain...Most common cause of hairiness is genetics, which doesn't impact pregnancy.
Birth or pregnant? Trouble giving birth? No. Trouble getting pregnant? Maybe, but usually not. Most hairy women are just hairy - due to race, genetics - but have no disorder. If periods are infrequent, then there may be an underlying disorder that can impact fertility.