If my boss asks me if I can help him to do something,
I reply: I'll bend over backwards to do it.
Does this reply literally have a meaning of flattery?
Learn English – Does “bend over backwards” have bad meaning
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None of Webster, ODO, Collins, or MacMillan lists "At top speed; frantically" as a possible definition of "head over heels". They only note the two definitions that the OP is familiar with. However, I can imagine without any great difficulty that frantic top-speed running could well involve chaotic stumbling and tumbling, and vice versa. So, it's not all that much of a stretch.
He looked scared, and that was no wonder, because head over heels after him came five other boys.
This could easily have been written by Stephen Fry with wholly different connotations.
Here is the entry from "break a leg" in Robert Hendrickson, The Facts on File Dictionary of Word and Phrase Origins (1997):
break a leg! Break a leg! means "good luck" in theatrical circles, probably not because the great Sarah Bernhradt "had one leg an it would be good luck to be like her." No one is sure, but one theory has the expression deriving, possibly through Yiddish, from a German expression meaning "May you break your neck and your leg," for which I can find no satisfactory explanation. It may also have something to do with wishing someone a "big break," that is, good luck leading to success. Or bad luck like breaking a leg a leg might simply be wished because actors, a superstitious lot, have long believed that wishing them good luck guarantees something terrible will happen.
Nigel Rees, Cassell's Dictionary of Word and Phrase Origins (2002) has this take on the phrase:
break a leg! A traditional theatrical greeting given before a performance, especially a first night, because it is considered bad luck to wish anyone 'good luck' directly. Another version is snap a wrist! Partridge/[Dictionary of] Slang [and Unconventional English] has 'to break a leg' as 'to give birth to a bastard', dating from the seventeenth century, but that is probably unconnected. As also is the fact that John Wilkes Booth, an actor, broke his leg after assassinating President Lincoln in a theatre. The Morris Dictionary of Word and Phrase Origins (1977) has it based on a German good luck expression, Hals und Beinbruch {May you break your neck and your leg). Perhaps this entered theatrical speech (like several other expressions) from Yiddish.
Other theatrical good-luck expressions include merde! {French: 'shit'}, toy! toy! and in bocca al lupo {Italian: 'into the wolf's jaws'}, although this last has been heard in the form 'bocc' al lupo'.
And Christine Ammer, The Facts on File Dictionary of Clichés, second edition (2006), has this:
break a leg Good luck. There is some dispute as to when and where this seemingly nasty advice originated. It may be a translation of the German Hals und Beinbruch {Break your neck and leg), which allegedly originated among World War I aviators jocularly wishing each other well. In any event it became widespread in the theater, both in Germany and later in the United States, and then came into more general use. It still is most often addressed to performers of some kind.
All of this comes as something of a shock to me, since I was assured by a very trustworthy fourth-grade teacher, just before our school orchestra performed a recital on the stage at the far end of the lunchroom, that "legs" was theater talk for the pulley ropes that are used to slide the stage curtains open and shut, and that urging someone to "break a leg" was really encouragement to be so good that the curtains would be closed and opened multiple times for encores—so many times and so vigorously, in fact, that the ropes would break. Therefore, "breaking a leg" indicated achieving a theatrical triumph. (Something similar to this theory appears in the "Non-literal" subsection of the Wikipedia article on "break a leg.")
I have heard the expression "break a leg" all my conscious life (from the 1960s forward), and I believe that it is extremely well known today in the United States—hence its inclusion in a book of clichés. I do not know, however, how many people who use it are laboring under delusions of its supposedly benign meaning, foisted off on them by duplicitous elementary school teachers.
Best Answer
In my mind, it's a very extreme form of commitment reserved for out of the ordinary scenarios. Not something I would recommend telling a boss on a regular basis, as it robs the term of its degree of commitment.
You might bend over backwards to win a key client from a competitor, but you wouldn't bend over backwards to get the weekly report in on time.
Even then, to use the phrase is indicative that you really need something to happen and that the consequences for your job/reputation/well being might be extreme if you couldn't. One typically bends over backwards not just for someone in a higher position than oneself, but also when the person you're doing it for shows little regard for you, because they know they're in control due to their station.
People who bend over backwards often assume a submissive role, which is not to be mistaken with there mere seniority that a boss would have over you at work.