Your questions are interesting, because while in principle all the things you are asking about are allowed, in this piece they look like errors because the composer doesn't understand/care how to spell the chords he is using, making it confusing for the reader.
1. The Db followed by C# two measures later. Should you avoid using the same note but different notation?
So to reiterate what I said, it's not wrong to follow up a Db with a C# if that's what the theory dictates to spell the harmony you are using correctly. In this case I would use a Db instead of the C# because it fits better with the harmony I perceive is going on in those bars.
2. G# and A# instead of A and B. Isn't it better to use the note of the main key?
Not necessarily, for example, if you have modulated to B major (which it is what it looks like they have done) it makes sense to spell things in B major. Unfortunately, the spelling of the right hand gave me a fatal disease, it could just be spelled in B major and related keys too and the whole passage would make more sense.
3. The piece seem to have sharp, flat and natural all over the place. It's giving me aneurysm reading it. Shouldn't you need to keep accidentals to a minimum and keep it one direction preferbly the same as the main key (in this case keeps mostly flat)? I would rather the score make it easier to read first, any type of music theory or mode change should be outside the score and analysed later.
Accidentals following music theory would make this easier to read. The reason it's so messy is because a lot of accidentals have been arbitrarily chosen rather than based on theory, at least to my eye. For example, if you are used to reading in B major, if everything in example 2 was spelled in B major it would help you read it because you'd recognize the key. Now that they are mixing sharps and flats and naturals unnecessarily it becomes confusing to read.