Yes, it is possible. It is a way of intensifying the idea of "I." "I did this. The fact that it was me is important. I did it--not you, not he, not anybody else, but I, myself." It can be used when the speaker says something general, and then points out that he is an example himself. It can be used when the speaker is talking about something other people have done, then pointing out a contrast with himself. "There's nothing wrong with putting ketchup on scrambled eggs. Many perfectly good people put ketchup on scrambled eggs. I myself put ketchup in scrambled eggs." Doing a web search for examples, I find: "None of us are perfect. I myself am peculiarly susceptible to draughts."--Oscar Wilde. He is moving from the general (none of us are perfect) to himself as an example (pointing out a way in which he is not perfect). "He was one of the few men of letters who really appreciated Oscar Wilde, though he did not rally fiercely to Wilde's side until the world deserted Oscar in his ruin. I myself was present at a curious meeting between the two..."--George Bernard Shaw. Here, he is just stressing that he is talking about the meeting at first hand; he had been there himself, he hadn't just read about it or heard about it. "I was... still haunted with the shadow of something she had not told me. I myself had kept back nothing, but there was a word Mrs. Grose had kept back."--Henry James.

Yes, you could say that when expressing a personal opinion or like/dislike. For example, "I myself don't really enjoy traveling, but my sister loves it!" I think it would be an odd thing to see in written communication. It may not be proper grammar. But it is definitely used in spoken English. For a more formal way of expressing the same idea, you could say "Personally, I don't really enjoy traveling, but my sister loves it!"

It just sounds odd and unnecessarily redundant. I honestly can't think of an instance where you would start a sentence with "I myself" and it not sound strange....