I am a university teacher and I never allow students to have sentences starting with conjunctions like and/but/or. It looks weird when used together with formal academic phrases. It actually seems to me that academic writing is getting more restrictive by the day--I've wondered if it's due to English becoming global, and foreigners, used to a more formal structure in their own language, writing papers and theses in English. Academic writing is off on a planet of its own, really. In creative writing it works sometimes, but I don't often feel the need to do it. I think it gives the writing a disjointed and flippant feel if it's used too often.
The bolded part of the sentence should become a mantra. It applies to so much of writing! The difficulty is deciding what constitutes too much. Is it carefully chosen for effect, or is it a habit?
This is such a simple rule that even Microsoft will ding you for it. Here is my rule about rules: if you don't know them and break them, it's a mistake. On the other hand, if you know the rule and break it on purpose, it can be cool. But why distract the reader with poor grammar when good writing goes much further?
Why implement a sense of style in your writing when you can blandly tell a story like a series of facts? I remember being taught in second grade that it was okay to begin a sentence with "but" and "and" if used selectively. If you asked me what "selectively" means, I'd say, when using it helps the flow of the paragraph or passage as a whole.
Microsoft Word has no sense of judgment. Neither does ANY piece of software. In fact, there is no softwqare that does an adequate job of analyzing the grammar of an English sentence, and it's because of the nature of the language. In technical terms, natural languages like English are what are known as context-sensitive grammars, which are inherently not guaranteed to parse unambiguously. You cannot truly parse English syntax without understanding the sentence semantics (meaning). But a sentence need not be well-formed to be good writing. Really! And I just gave three examples in this paragraph of ill-formed sentences that are acceptable writing.