If I eat fatty foods, will my baby gain extra weight from my breast milk?

No. No - your body will make the breast milk composition its meant to make as long as your body has enough of the basics. As long as you eat a balanced diet, you should be fine. In essence, you should feel free to eat or drnik anything you wish (with large restrictions to things like alcohol of course) and your baby will do well. Breastfeed is great - but enjoy your food as well!
No. No. There is foremilk and hindmilk. The foremilk has protein and glucose. The hindmilk has more of the fat and satiates the baby. The longer you breast feed the more fat the baby will have from one side of the breast. This is not related to how much fat you intake in your diet, but rather to how long they breast feed per side.