Is there any requirement to wear a hat or jacket while walking on the street?
(aside for style)
It depends: If you are a member of a Yeshivah or community in which for a person of your spiritual stature it is considered the uniform then you should wear it. If it is considered the appropriate respectful uniform (for you) in your community then by not wearing it (unless it is too hot) you are making a statement of making arbitrary change which could be considered a. a lack of Tniyus (demureness) and b. poreitz gederI , ripping down the fences that define a good society. But, once again, that is only if it is not too hot.
Suffice it to say that if you find yourself in a community where men do not wear a hat for davening, there are those who would say it is inappropriate to wear it (See "Collected Writings" or "Collected Speeches" of Rav Shimon Schwalb. I forget which.)
If a boy is a member of a Yeshivah that requires its students to dress so in the street, he should do so, or find a yeshivah he is more comfortable with.
The Ketzos Hashulchan writes that "the way of modest people is to wear a hat on top of their Yarmulke".
His source is the siddur of the Yaavetz.
On a halachic level, the Rambam says in Hilchos Tefillah 5:5 that one should dress for davening as if he were in front of a "gadol", presumably a King of sorts. The Mishnah Brurah adds in 91:5 footnote 12 that one should wear a hat like they do "nowadays" in the streets.
The presumption nowadays I believe is that no-one walks in the streets with hats, nor do they greet people of high stature with hats on. This is known to be attributed to John F. Kennedy who made it customary to remove a hat as a sign of respect.
Some believe that it may depend on the place you live. I asked a prominent Rav for example this question and he told me he believed that in the USA it is still customary to wear a hat to show respect. If, he said, he were to move to Israel, he did not see it necessary to wear one.
Some say that nowadays there is no such thing as a "gadol". The fact that "prominent" people do not require/show respect through a hat is only because they try to relate to the common man who wishes not to. That, or the fact that they are not truly "gadol", could be reasons as to why the public would not see it necessary to wear a hat, whereas in truth it is a lacking in them not to when davening. This theory of course only applies if the main reason for wearing a hat is based on standing in front of a gadol. If we base it on how people dress in the street (as the mishnah brurah adds), the reason for such a custom is almost irrelevant. As long as they don't, one needn't either. However, I do not believe he referred to wearing one all the time, but only at a time of prayer.
Other reasons for wearing a hat may include a minhag (custom) passed down through one's family to wear one. One may also just wear one to relate and "fit in" with a certain "group" (if it is not the full-fledged custom of the place of course). One should be aware in such a circumstance that it is not a halachic matter, but rather a stringency on themselves to do so, otherwise it may cause unfortunate misconceptions of himself and others. This may be a reason for why one wears one more than just while davening.
I have not been privileged to understand, however, why there may be a custom to wear a hat everywhere. For instance, in a pizza shop or at a wedding, I cannot seem to understand why such a thing would require a hat. Perhaps some extend the meaning of a hat to all services of Hashem (the bracha at a pizza shop), although it would seem that the simple reading of the sources are speaking only about when one is davening.