Question: When Travis goes into his fitness phase, it shows him hovering his fist over the stove fire. Why is he doing this? What does that achieve?
Question: Why was Travis labeled a hero at the end and not arrested? He murdered several people.
Chosen answer: One theory about the end of the film is that it is Travis' dying thoughts, but this is not the view of Scorsese or writer Paul Schrader...they intended the ending to be ambiguous and an ironic critique of the media's, and the public's, reaction to and interpretation of violence (Travis is hailed as a hero for rescuing Iris, but we can imagine a very different reaction had he followed his original plan of assassinating a senator). Interestingly, when the film was originally shown on television, the following "disclaimer" of sorts accompanied the closing credits: "In the aftermath of violence, the distinction between hero and villain is sometimes a matter of interpretation or misinterpretation of facts. 'Taxi Driver' suggests that tragic errors can be made. The Filmmakers."
Question: At the end when Betsy is in the cab, it seems like Travis says hello ages after she got in. So it seems like that scene is imagined. Is it?
Chosen answer: No, it just seems that Travis had no idea what to say to Betsy and after a while, a simple hello is all he could manage.
Question: Is there an official explanation to the ending of the movie? Some people say that it was Travis's dream sequence, and others say it was Travis going back to his manic depressive self again. What caused Travis to get so startled when he was looking at the rear vision mirror? Did Scorsese deliberately make the ending very vague or is there supposed to be an explanation to the ending?
Answer: This is just my opinion. Remember Iris's line "Have you ever tried looking at your own eyeballs in the mirror?" or something to that effect. Well, Travis sees the madness reflected in his own eyes, doesn't like what he sees (as it reminds him of what he has done, what he might yet do).
Answer: There is. Both Scorsese (in the audio commentary) and screenwriter Paul Schrader in multiple interviews establish that the ending is not to be taken as a dream sequence or anything of the sort. I love the previous answer, by the way, it does nail what has also been stated; while Travis survived this time, it's very likely there will be a next.
Question: When Travis first entered the porn theater, why was it blurred? Was that Martin Scorcese blurring the picture to avoid an NC-17 rating, or was the porn movie supposed to be blurry?
Chosen answer: It could be for several reasons. As you pointed out, it could be to avoid an NC-17 rating, though Scorcese could easily have used scenes that were not specifically pornographic. It may also be to keep the audience's attention focused on Travis and what was is going on in the scene and not be distracted by any pornographic material. Also, if you are referring to when the movie was being aired on TV, then any pornographic materials would have been blurred out to comply with FCC standards. Nudity and pornography cannot be shown on general TV stations.
Question: Why did Travis want to kill Sen. Charles Palantine?
Answer: To get the attention of Cybil Shephard's character.
Answer: It helps him create a tolerance for pain. As we see, he certainly gets a lot of that dished out to him.