I don't think anything is as misunderstood as sentence fragments. I'm not sure I've ever seen an actual sentence fragment used in a published work.
The trouble with rules of grammar is that they often apply to a specific case, and then everyone tries to apply this case to something else.
The rule is about what is and isn't a sentence fragment is very well defined for a standalone sentence.
The standalone rule would make half the sentences we read fragments, but they aren't.
Many grammar books stop right there, and few read the ones that do not. I think I was in year three before an English professor actually spelled this out for us, though I think we all knew it on an instinctive level.
If a sentence is given complete context and meaning by the previous sentence, it is not a true fragment. It's next to impossible to fit a true sentence fragment into a work, and I'd say it's always done because the writer simply can't write well, and knows nothing at all about grammar.
As for ending a sentence with a preposition, nine times out of a ten, it's a very, very bad idea. Doing so one time out of ten can, however, make a sentence shine. The trick is knowing which time is the right one.