I think that you are going to get a great variance of responses on this type of question. I would initially say that Shakespeare's lack of ego is what makes him uniquely different from the Romantic thinkers. Part of this is that Shakespeare did not so strongly identify himself as part of a movement or an intellectual community as the Romantic thinkers did. Many of the Romantics saw themselves as innovators and cutting edge thinkers because they were with others who saw them in much of the same light. Wordsworth and Coleridge both saw themselves as intellectual pioneers when they compose their arguments behind Romanticism in their works such as "Lyrical Ballads." Shelley identified himself with Byron so much that both fed off of the other in their desire to claim immortality. Yet, Shakespeare seemed to operate alone. At best, one might be able to claim that contemporaries like Cervantes might have stood along side him, but Shakespeare did not see himself as part of a community like the Romantics did and I think this might be a reason why he did not assert his greatness as the Romantics would have had they developed much of what he had done. The profound nature of works like "Macbeth," "Hamlet," or "Othello" would have made the Romantics gush in praise and adulation, but Shakespeare did not seek to consciously assert his own greatness as the Romantics might have done. I think he carried himself with a great deal of confidence about his work and how it should be staged, but I don't see him as one that sought to eagerly claim his rightful place in the pantheon of literary greats as the Romantics did later on down the road.
I agree with the previous post that you're likely to get varied responses to this question--which is quite an interesting one, by the way. It seems to me that Shakespeare did not see himself as any particular kind of genius. He was a master at taking stories (or pieces of stories), legends, events from history, and common superstitions and beliefs and weaving them into a story appealing to both royalty and the common man. The two tragedies you mention, Hamlet and Macbeth, incorporate most of those elements, making them particularly appealing to the Elizabethan audience (for example, the use of ghosts and witches and historical context). His genius, I think, was writing what his audiences liked while demonstrating the literary and language skills which have been admired for centuries. There is little evidence either that Shakespeare was part of any real writing community or that he set himself apart as a genius. In contrast, the Romantics, it seems to me, were enamored of themselves and their work and used every opportunity to exploit themselves.
I think one key difference between Shakespeare and the Romantic writers, especially the poets, is that Shakespeare was crafting a drama and there would be little to no place for an "autobiographical" voice in the plays. It is a bit of leap to say that because Hamlet contemplates revenge, that Shakespeare believes revenge is good or bad. The poets, on the other hand, do have poems that more clearly represent their own personal thoughts and perceptions of the themselves, what they believe in, and how they view the world around them. Shakespeare's play do represent attitudes of his time, but what evidence do we have that the plays represent the ego of the writer?
I would have to agree with other posters above - whilst I personally think that Shakespeare was indeed a genius, I am not too sure that he would have had much opportunity to get that much of an ego in terms of his literary profession. One does get the sense that the Romantics were rather egotistical, which you don't seem to get from Shakespeare. Certainly his skill was taking tales and re-writing them, improving them and contextualising them for his times, and indeed, establishing universal themes, which make him a genius and an extremely talented individual.
There's no way to tell, since we know so little about Shakespeare or even if he was one person. However I do think he was proud of his cleverness. People who like puns usually are. I am not sure if he thought himself an actual genius though.
I agree with answer 5 - it's a little iffy, trying to glean Shakespeare's thoughts when we have so little to go on. To try to guess at his thoughts about himself is even sketchier.
There may be hints scattered about, but in the end we just can't know, and I don't think there's enough hints to warrant a guess.
Shakespeare does not see himself as a genius, but a man of great insight into human character and creating great dramatic stories that anyone can follow on so many different levels. His characters in the plays you mentioned have flaws that bring down their characters to the worst possible position, death notwithstanding.