One of my colleagues asked me the following question:
Do you happen to know when the ‘rule’ of not beginning a sentence with a conjunction was formulated?
This is just the sort of question that we would be able to answer once our database of usage guides and usage problems is up and running, but it isn’t as yet, so does anyone else know? Why was it formulated to begin with? And do you object to starting a sentence with and or but, or any other conjunctions? If so why?
Bridging the Unbridgeable 
Type “starting a sentence with a conjunction” in Google and you get lots of interesting results.
The Fowler brothers deal with sentences starting with “but”, “and”, and “for” extensively in The King’s English (1908), but in the context of a conjunction followed by a parenthesis: they do not comment on the initial buts and ands.
” But with their triumph over the revolt, Cranmer and his colleagues advanced yet more boldly.—J. R. Green.
The adverbial phrase is with their triumph over the revolt. But does not belong to it, but to the whole sentence. The writer has no defence whatever as against the logician; nevertheless, his reader will be grateful to him. The familiar intrusion of a comma after initial And and For where there is no intervening clause to justify it, of which we gave examples when we spoke of overstopping, comes probably by false analogy from the unpleasant pause that rigid punctuation has made common in sentences of this type. ”
In “Modern English Usage”, there is still no mention of this issue. Only in the 2nd edition, in the ’60s, is the use of “and” to start a sentence explicitly condoned by Gowers, the editor.
This might be evidence that it only became an issue sometime during the Interbellum, or in the decades before and after.
I’m afraid I can answer only one of your questions, so for all it’s worth, I do use sentences such as: “And another thing”