Can you start a sentence with the word And?

Answer 1

Yes. But be careful with it, because your writing can get that run-on sentence feeling if you do it too often. Usually the word AND will be used to continue a thought from the preceding sentence.

Some old-school grammarians still preach against it, but usage-wise, many good writers have done it, and it is perfectly acceptable. The most important thing is to avoid overuse. Make sure there is a purpose behind the choice and not just ignorance of sentence structure. If you keep those things in mind, it can be done well and to good effect.

Answer 2

Yes. There is no word or phrase in English that cannot begin or end a sentence.

Although grammar mavens and grammar marms and old-fashioned English teachers will say that, no, in formal writing you should not begin a sentence with "and," in fact sentences are started with "and" all the time (especially in informal writing and in speech). Moreover, many of the very best writers in English historically do occasionally start sentences with "and"--from Shakespeare to Yeats to Toni Morrison. You have to sit back and judge who your target audience is: if you're writing for an old-fashioned English teacher or for a very formal purpose, then avoid starting sentences with "and." Otherwise, go by your ear and what you notice other good writers of English doing when you read material by them.
NO