1. Yes, there is much interesting philosophical work on modern art. I would start with Arthur Danto, who has written many interesting essays (often for the Nation) and a few fascinating books on the topic.
2. The term aesthetics is certainly applied to music -- see Theodor Adorno and currently Lydia Goehr and Peter Kivy for example -- as well as to poetry. Less frequently to literature, but this is presumably because there aesthetic quality is typically a less important component of overall quality (esp. outside fiction).
3. "Logic" is perhaps not quite the right word for what you have in mind here. Perhaps "rules"? I would think that aesthetic judgments are intuitive judgments, and that any rules laid down for composition or appreciation have standing only insofar as they are confirmed by intuitive judgments. (Intuitive judgments may differ, as they did in respect to the atonal works of Arnold Schönberg, for instance, and judgments about rules will then differ correspondingly.) To what extent can our intuitive aesthetic judgments be expressed in rules? This question has been quite interestingly addressed in modern times by reference to the apparently quite fundamental distinction between objects that are and objects that are not works of art. Is there a rule for drawing this distinction? Marcel Duchamp raised this question dramatically when, in 1917, he took an ordinary white gentlemen's urinal, called it Fountain, signed it, and put it on display. In 1962, Andy Warhol began displaying Campbell soup cans. The debate about what is art, and what is good art, is ongoing; and Arthur Danto's work offers an excellent entry into this debate.