#4 Zum Thema badly als einzig grammatikalisch korrekte Form... Außer in Phrasen wie "I want it so badly", würde ich das badly eher als "auf eine schlechte Weise" interpretieren. Hier verwendet klänge es so, so als gäbe es auch eine gute Weise, wie das Bein schmerzen könnte
Technically, hurt is always followed by an adverb, because it's not one of the link verbs (eg taste, look, smell, feel). Replace 'bad' with any other adjective and you'll see that only an adverb is acceptable:
My leg is hurting
My leg is hurting
My head hurts
So hurt needs to be followed by an adverb. I don't think this has anything to do with the interpretation of the sentence, and whether 'hurt bad' means something essentially different to 'want badly'. The question is: can bad be used adverbially?
As I said in #2, opinions will differ on that. I'm not a grammar prescriptivist, I accept that words change over time, and I don't have a problem with the fact that one day 'bad' might be considered 'standard'. I don't think it's there yet, though, certainly not in the UK, and it sounds very sloppy to my ears. It would be rare in my circle of friends, and it would be marked incorrect at my girls' primary school.
The UK dictionaries list it as informal / North American English, informal / 'Many people consider this use incorrect' / not standard / Now colloquial (nonstandard) (chiefly North American in later use).
AE speakers will have to comment on whether the adverb 'bad' has become mainstream or whether it's still a touchy subject.
(*note that neither of the sentences with lightly or strongly sounds especially idiomatic; I think English speakers would tend towards different ways of expressing this, probably with 'badly', eg My leg isn't hurting too badly. I was just trying to illustrate that hurt needs to be followed by an adverb, and these were the best adverbs I could come up with.)