Caulking a round corner stand up shower
I have dealt with a lot of tub/shower surrounds.  100% silicone caulk (GE II is best at big box).  Most install instructions for the surrounds will actually tell you 100% silicone - some will give you their preferred brand.  

Answers:


You don't need to take it apart.  This may make it easier though.  Weigh how much time you will spend taking it apart and putting it back in.  
All of the old caulk needs to go and you need to be caulking on a very clean surface.
It should be sitting flush.  Why wouldn't it?  The fact that it isn't sitting flush tells me that it is moving (even if very slightly).  There is no way this 20 pound shower door is staying perfectly still suspended at certain points.  This movement will cause constant issues with your caulk and the fact that it will cause the issues in the easiest place for water to get in is a double whammy.  It isn't hard to install these things right.  There is no reason this should be sitting right (if it is bent then it needs to be bent back).


Other notes:


Sorry to say but this whole job looks hack to me.  I am not sure about the "main" issue (other than the people working on it) but it looks like maybe the wall isn't perpendicular or base isn't flat.  
On the last picture it looks like you are actually missing a trim piece on the inside.  You must have something there to salvage this shower. I would simply call the manufacturer.  Ask them if they make something or have replacement parts for your shower.  You might want to order a new enclosure.  Your actual shower looks fine it is just the door system so I don't think this is a tear down for sure.
The caulking looks ridiculous.  This is "one-month" caulking.  More caulk isn't better.  The more caulk you apply the greater chance of the caulk being moved, bumped, whatever.  Then the seal on the caulk will break and a .5 mm crack can allow tons of water to leak.  Also all of your caulking should be from the inside of the shower.  Anything on the other side is cosmetic.