Both of these sentences are correct, and they carry same meaning. Depending on the setting, however, one may be preferred over the other, and which one that is will change, again, depending on the setting.

In this instance, "that" is showing causality. Put another way: "the following year's unrest" was partly caused by "the energetic debate among intellectuals about how to reform China", which was, in turn, "prompted" by "The documentary programmes" The word "that", therefore, could be put between "The documentary programmes" and "prompted the energetic debate among intellectuals about how to reform China", replacing the word "had", and taken out of where it is in the original sentence, without changing it's meaning: "The documentary programmes that prompted the energetic debate among intellectuals about how to reform China helped foment the following year's unrest."

It literally means "She is the bread of God", but how that might be used in English depends on the context. Presumably, this is meant as a compliment to a woman, but what is it about her that it is complimenting?

It means that the popularity of car sharing has greatly increased. The term "takes off" usually refers to the departure of aircraft, but when the context is about other things, it refers to getting started or becoming more commonplace.

I would say either, but in both instances, there are more than one. There are at least a half-dozen American accents, and there are dozens of accents in England alone (not including the three distinct accents in Wales, the nine or so in Scotland, or the accent predominant in Northern Ireland). In just the county of Yorkshire alone, there are 3 different accents, and in Greater London, there are several accents that are broadly distinct from one another.

It can mean a couple of things: Within a debate or argument, it means that the speaker is conceding the validity of a given point, that the point is reasonable (fair) enough to not warrant arguing against. In most other contexts, it is usually a statement made to show that the speaker views some act, circumstance or comment as being just and equitable (fair) In a far less common context, the word fair means pale... this context is very specific, and would likely be found only in literature of previous centuries: "His skin was fair enough, that he could walk the streets without being instantly identified as a runaway slave."

That line is an example of what is known as "poetic license". In poetic license, the rules of the language have been deliberately suspended for the sake of obtaining an effect. Although I do not know of this song, I suspect the character that the song is referring to doesn't care about adhering to rules, and so the use of poetic license would be entirely in keeping with that. In that case, while it is not correct, it is appropriate.

As addenda to Diego's excellent answer, I have a couple of points. The Determinate Article is also (and possibly better) known as the Definite Article, and the Indeterminate Article as the Indefinite Article. If you see the term "Definite Article" or "Indefinite Article", they still refer to "the" and "a"/"an" respectively. A note on choosing between the options in the Indefinite Article: "An" is used when the following word (whether it is an adjective or a countable noun) starts with "hard" vowel, i.e. A, E, I, O, or U "A" is used when the following word (again, whether it is an adjective or a countable noun) starts with a consonant. The choice is based around how the word is spoken, not how it is written. This is a particularly important point, because there are words that would appear to have the wrong indefinite article in front of it when written on a page. In fact, the choice of "a" or "an" provides a clue on how to pronounce the following word. There are some words that start with a vowel which are correctly preceded by "a" instead of "an", for example "Having graduated high school with good grades, he felt ready to find a university to attend." This is because some words starting with "u" use the long u ("yoo"), which is considered a consonant, and therefore warrants "a" rather than "an". There are also some words that start with a consonant which are correctly preceded by "an" instead of "a". The most common example of this is the word "hour"... in this example, the "h" is entirely silent. The first voiced letter is a vowel, and therefore, the word receives the article "an" in cases where the article "the" or a number is not used. This is not true of all words that start with a silent letter, only when all the consonants before a voiced vowel are silent.