If I eat fish, can the baby get mercury poisoning?

Yes. Many clinical studies have found that mercury is transferred to infants by nursing. That may not result in "poisoning" in the technical sense, but no level of mercury is truly safe for the developing infant brain! Mercury has a devastating effect on developing brain. Search "minamata"by photographer, eugene smith, who documented the mercury poisoning of japanese village in 1950's - disturbing pics.
No. There have been lots of studies on this topic and so far none have found an association between the fish mom eats and any risk of poisoning in her baby. Some groups do recommend, however, limiting canned tuna to 4 ounces a week to minimize mercury exposure for mom.
No. Not unless a breastfeeding mom has high levels of mercury, should a baby be able to get mercury poisoning just from breastmilk. The concentration of mercury in the breastmilk is less than the level in the mom's blood. Breastfeeding moms can limit their fish intake to 2 meals a week, and avoid the high-mercury types of fish, just to be safe. Mom's who eat much fish can get a blood mercury test.
No.. ..A problem. Smaller fish (salmon, flounder, etc) are only at risk if caught in a high risk area which is actually illegal.
No. There have been numerous studies in the medical literature trying to establish a relationship between mercury consumed in fish by breast feeding mothers and poor intellectual outcomes in breast-fed children and to date, there is no association. The problem isn't going anywhere however, and future studies may show an association. For now, though, the answer is no.