I would leave the rudder as long as it isn't leaking. I would pull the shaft though.
Like Salty said, you can go either way. I removed my prop and pulled the shaft out the top with the coupler still instact..
I haven't gotten mine out yet. I messed with it for a while and kinda gave up. Once I "need" to get it out I will. Mine leaks some, so I want to fix it before I put the boat completely back together. It looks like I may need to use some sort of puller. That seems to be the general concensus.
Measure and make drawings. Use the old sections you cut out as a reference. I'm taking out and putting back one stringer at a time. Right now I'm working on the port side and the starboard side is still in. I've found I've gone over and measured there a few times already just to check a couple things I didn't really think to measure in the first place.
Salty covered this. You'll re-use those when you put it back together and cover it with the rubrail.