Okay, a little history here (based on our previous discussions). While the dropped quote method has been used in traditionally published books for a long, long time across the world, it seems the only authors/readers familiar with the rule are Americans (probably because it's actively taught in English courses). While Europeans aren't quite as judgmental, agreeing that it's widely used, our Australian contingent—for whatever reason—is adamant in their opposition to the practice, even after conceding it's been used in many of their older classics. Again, it's mostly a matter of what authors and readers remember being taught in school, rather than what they've been exposed to for the past two hundred years.
Ernest has a lot to say on the subject, as he not only refuses to use it, but goes to great lengths to avoid the situation entirely. Whenever a speech by a single speaker spans multiple chapters, he inserts a new action attribution (ex: "Tom stood, confronting the others." instead of "Tom said,").
While that's perfectly valid, it IS an incredible amount of extra work for a lengthy story.
In my own case, since I rely heavily on dialogue, I have MANY long monalogues. In the latest chapter in a story I'm still working on (just wrote it last night), the MC goes on a rant, and the non-breaking lecture emphasizes just how angry he is, not allowing anyone to contradict or even react to what he's saying until he runs out of steam. His tirade spans seven paragraphs, chewing up 677 (pre-edited) words, but it gets the point across that the man who's been holding everything in for his entire life has finally reached the end of his rope (i.e. I have no intention to break his monologue up with minor action breaks).
You can either use it, ignore it completely or try Ernest's technique, as it really doesn't matter, but trust me, there's a LONG historical tradition behind this technique, even if few non-Americans recall seeing it.
As for starting a sentence with "And" or "But", the traditional guidelines (mainly written for scholarly non-fiction), mainly focused on those two words. If you follow that advice, "But" is easily replaced with either "Yet" or "However,". Technically "And" should always be appended to the previous sentence, but for fictional dialogue, since it's how most people speak in their daily lives, it's fine (though I'd be cautious about using it in the narrative).
As others have noticed, I'd also avoid starting a paragraph with either word. Normally, when you do use them, it's a continuation of the previous thought/sentence. However, when starting from scratch, it sounds especially jarring to most readers.
I've avoided using the convention on SOL for that very reason, plus some readers are unfamiliar with it. But there's a third reason too - monologues are B O R I N G!
Sorry, Awnlee, but for me, my dialogues are my main strength (especially since I'm SO bad at SHOWing without it). In fact, many of my readers' most favorite chapters are the long and involved discussions by the various characters that typically follow a big fight, as they debate what actually happened and ends with the main character providing his interpretation of events.
It isn't the monologue that's entertaining, but the verbal dynamics between the characters, which make the otherwise boring data dump of information entertaining while revealing many of the key elements driving the story. (And since there are many competing ideas being tossed around, it's easy to throw in multiple foreshadows and red herrings to keep them guessing about just where the story is heading.)