Fixing water damaged drywall

A while ago I had some water damage inside the house and I would really like to fix it

Is this something I can just patch up with drywall putty and paint over it, or do I need to cut out this section and put a patch panel in?

From the looks of it, they used latex paint
Can i cut the piece of paint that had water behind it, sand the edges and repaint it?


And how do recreate the texture on the ceiling?

Thanks guys