The preposition "aan" is important here, to distinguish between the two meanings. In the sense of introducing yourself, "zich voorstellen" is an intransitive verb (it doesn't have a direct object), but it can have an indirect object with "aan" (the person you are introducing yourself to). On the other hand, one would always need to imagine something, you can't just be imagining.
- Ik stel me voor. - I introduce myself.
- Ik stel me aan iemand voor. - I introduce myself to someone.
- Ik stel me iemand voor. - I imagine someone.
Those examples are very useful! Thanks! W.r.t. objects, in the first and second examples, the reflexive "me" is acting as direct object ("aan iemand" is a prepositional phrase with "iemand " as its object). In the third example, "me" is, as you say, an indirect object meaning "to myself", while "iemand", the receiver of the act of imagining, is the direct object.
From an actual linguistic point of view it's not sloppy English to put a preposition at the end of a sentence - it's a "rule" imported by "purists" from Latin that has been proven to be incorrect MANY times. It's often more sloppy to try and move prepositions to other positions, as it can make sentences sound more awkward.
I am not a native English speaker, so forgive if I am wrong. You are correct about the location of preposition, but I think "to who" is just wrong English and it should be "to whom". So even if you put "to" at the end of the sentence, I think it should be "Whom do you introduce yourself to." No?
No, you are correct. But nowadays, 'whom' is almost always considered to be archaic. You will rarely encounter an English speaker who will actually use 'whom'. It's acceptable to use 'who' in all contexts, even in formal speech and writing (although, there may be some who still prefer to use 'whom' in formal speech).
These days it's widespread; the horse is already out of the barn with "whom". Even more so with the preposition problem. (Even more so with the rhetorical use of fragments. ;) )
However, I agree that pedants shouldn't be penalized, as although "whom" might not be as popular anymore, it's most certainly still correct. I've reported it often.
Churchill was being facetious. That is one of those awful latinate rules you learn in school, but are honored in the breach in real life. English has tons of seperable phrasal verbs, the second (and third, if they are three part phrasals) LOOK like prepositions. They aren't, really, they are part of the verb. He put it on - , she looked it up - You HAVE to put the "preposition" at the end in these sentences because the requirement to have the pronoun between the main part of the verb and the particles that follow is so strong.
In modern speech it is correct.
However in writing - especially formal writing - using who when it should be whom, and ending a sentence in a preposition makes writing sloppy.
An easy tip on who vs. whom: If the answer to the question is him/her - use whom. If the answer is he/she - use who.
(Native English speaker who uses formal writing daily.)