In formal writing, it still really isn't acceptable. Even starting sentences with "Also" should be avoided (although not necessarily as bad as starting with "And").
Starting a sentence with "And" is likely more common in speech or in a passage which is intended to imitate speech (or written in first or second person); the result of essentially substituting a comma (or semi-colon) for a period is then what results in the unwitting error, which in speech results in a firmer emphasis of the second statement / clause. For example:
"These documents must be at the client's office tomorrow by 8 am, sharp. And I really mean it this time!"
Of course, in the example, one could probably drop the 'And' and it will still makes sense, although there is a small sacrifice in carried meaning.
I use "Yours sincerely" sometimes but have avoided "Yours faithfully" for quite some time as it seems to (whether incorrectly or not) carry a somewhat religious connotation, or doesn't seem to convey a correct emotion for the letter which was written. More common usage is now "Best Regards" or "Kind Regards", but I tend to use "Yours sincerely" for those times when a written / signed letter is required.