What you're talking about is called playing rubato (https://en.wikipedia.org/wiki/Tempo_rubato), and I think that is part of the point here. The control that you get over the timing allows you to express yourself musically / perform the music with your own interpretation.

The instructions may be a bit unclear, but I think they are suggesting that it works better artistically if you play some parts evenly (the three-note sequences that repeat) and add your timing variations to the other parts. This creates a kind of a baseline of regularity that makes the variations stand out by contrast. (It also gives the feeling that you're capable of even, steady timing and the timing variations you do incorporate are there on purpose.)

Indeed! I think this is what makes this so fun. I can also highly recommend listening to various recordings of the moonlight sonata and notice how different it is depending on the precise timing the pianist uses.