I would definitely say the rules of writing apply to the situation. In formal writing one may want to stick to rules more strictly to avoid judgement, while in emails or even forums the writer should not feel required to constantly check their quick words.
My opinion.
Writing english has a more artistic touch to it than math. I think using "and", or "but" at the beginning of a sentence works all the time with conversation in a written story or essay. However using those in regular writing makes the written piece sound sloppy and disorganized.
I think conjunctions at the beginning of sentences are absolutely hideous. I believe they call them conjunctions for a reason, no? As for spelling and pronunciation of words..the american way annoys me. I prefer to stick to the origins of the language and therefore try my best to adhere to the British version however, those pesky Americanised words snake their way into my vocabulary and I always have to make a mental note to hate myself later.
There is not a hard and fast rule, other than convention, that says we shouldn't begin sentences with conjunctions. Younger students and less advanced older ones are warned against it so often because it makes for poor reading and choppy style if used improperly or too often. I tell my more advanced students that using the conjunction is fine, provided there aren't better ways to construct the sentence. Usually, there are. We work with semi-colons, colons, dashes, and the old fashioned period/comma and conjunction. On occasion, using one of these makes the sentence too clunky, so a conjunction begins the next sentence. Most of the time, I try to help them work through the sentence to avoid such a construction. But there is no harm in it, if it is a matter of style or necessity.
As to spoken grammar, I agree that our spoken tongue has gotten less formal. So long as people make agreement, I can live with that. Writing, however, is my art, and I will not have it butchered. It's one thing to understand that English is a living language, and as such, it changes as society does; but it's entirely something else to allow such abominations as b/c, L8R, and other slang into formal writing without a fight.