What's the big advantage with having a full frame? More picture? They are also typical worse at handling noise right?
Where did you here that they are worse at handling noise?
The sole, absolute, and single most important technical aspect of a photo camera is the sensor SIZE. A larger sensor size means that more light is captured for the SAME exposure (obvisouly, it's the entire light capturing system, meaning, the lens diameter must scale). Tough to understand (the relationships between all the relative and absolute factors involved), but it's physics.
This extra light actually reduces the noise in a photo (when scaled to the same given physical size). Of course, it all depends on the used technology, but this is how the current technology scales.
You can get just as much picture with a crop sensor - just take a couple steps back. Now you have a higher resolution image of basically the same thing.
Not really, you have an image with a slightly lower technical quality. (Though it's unlikely to see the difference between a FF and a APSC at low ISO.)
Also, don't full frame cameras have issues with the edge of the picture sometimes?
No. Lenses have that. Besides, after I apply lens correction in post-processing, I add some vignette.