Is "hello." a sentence?
Yes.
What do you want for dinner?
Vegetable curry.
Hello! You surprised me.
Since you define a sentence as needing a verb, please realize that a sentence does not always have to express its verb. Also note that full stops are necessary in written language only. One does not punctuate spoken sentences the same way we do written sentences.
Additionally, hello is a greeting, which has a special function. When you greet people using spoken English, do you say Hello or do you only say something that has a verb, such as I bring you my greeting, which is hello or perhaps I greet you hello? Do you see what I mean? We don't always speak in fully expressed sentences. And since writing sometimes seeks to copy or represent spoken language, there will be times when something we write does not have an expressed verb, yet it will still be a sentence.
"Hello." is a good example of that. And if you want to know where the verb is, you can consider "hello" to be short for I greet you hello.
Likewise, hello! can also be an exclamation of surprise. See definition 3.
Other exclamations include
Bingo!
Dang!
Hogwash.
To me, these are sentences. If you need a verb for them to be a sentence, you can think of the sentence as
"(I say) 'Hello!'"
"(I say) 'Hogwash'."
The second line of this answer is one word: yes.
I condider "yes" in that context to be a sentence. It is answering the question that is in the first line. If you need a verb for it to be a sentence, you can consider the sentence to be
Yes, I say.
Okay? (Are you "okay" with that?)
Recommended additional reading:
Sentence Well-formedness
The general form "I don't know [question word], but" is a common idiomatic expression. The overall intent is to indicate some measure of surprise at an unexpected or unusual situation.
I don't know how, but the cat got on top of the roof and now she can't get down.
I don't know why, but the mayor thought it was a good idea to hold a press conference during a thunderstorm.
I don't know exactly who, but somebody ate all the chocolate cake I had been saving.
I don't know when, but some time recently the clocks were all reset.
And so on. I would memorize and practice it like any other idiomatic expression, as a complete phrase, since the individual words are not as important.
Best Answer
It is perfectly all right to begin a sentence with a conjunction. It is a special form of emphasis, used to elevate a clause to a position of more influence and importance.
It can also be used as a summation of previous statements.
Oxford Dictionaries Online calls the rule against starting a sentence with a conjunction a "myth" and states that
In sum, this is just one of many shibboleths about English that you can safely ignore.