Fixing water damaged drywall

Used to do these repairs all the time. I will assume that you have already fixed what caused the problem in the first place. In your first photo, water got in behind the paint and softened it up. More water coming in means higher hydraulic pressure and a bulge starts to form, more water and it starts to look like a balloon. If you were lucky it burst out the bottom and self drained. Strip off the the lose paint, it should just peel. Use a sharp knife and score an area about twice the width of the damaged section. Just score thru the paint, not the skin of the board. Because this part is vertical, the water should not have caused much damage, if any. Remove loose edges and scuff with an 80 or 100 grit sanding block. Apply a thin coat of all purpose drywall mud (this has a glue in it and is why it is used for taping) as a base. You don't have to be perfect as a thin coat of drywall Topping is going over this. That coat will be the one you sand to perfection with 220 grit or higher. Prime the repair the next day and paint the day after that. Couple of coats and you won't even notice it. Feather it out to get the best result, or paint until you get to a corner. Quick answer to the texture is a Crows foot brush. That area is a little more complicated to fix. PM me if you want mt .02 on that.