I think literally 情怀 means a sentiment or mood. But I strongly suspect that here it refers to a sort of Chinese tradition or traditional practice.

Vague concept into vague concept. Inhereit as in to inhereit (and continue) a tradition.

Beautiful, magnificent...whatever your preferred interpretation of 瑰丽唯美 is.

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Note added at 1 hr (2014-01-10 23:18:24 GMT)
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I mean "ethos", of course. Need to sleep...

As suggested

to (continuously) incite and further your longing for China with (the display of) her exquisite beauty and splendor
延续(您-English speaking audience对）瑰丽唯美的中国（的）情怀
情怀-longing

供参考

I think you should understand yanxu to mean "be an example of", and qinghuai as a dramatic/mythic tradition.

I assume the subject of the verb is the production?

XXX is a magnificent drama in the classic Chinese tradition
XXX captures the beauty and romance of the Chinese dramatic tradition
XXX is a classic example of Chinese drama, gorgeous and melodramatic

情怀means strong sentiment, feeling or thought, so it can be translated into a number of things and <expression of passion> seems to be the fitting one in this context. 瑰丽and唯美are both adjectives here, and although sometimes it is necessary to translate adjectives into nouns, here they modify中国情怀, and should be treated as so. 瑰丽can be translated as <magnificent>, <spectacular> or <visually stunning>. 唯美does not just mean <beautiful> or <aesthetically pleasing> but also <romantically inclined>. 延续means <to extend> and when in this context of 也将在舞台上延续, it clearly means <and will extend...to the stage>.

That would be the faithful translation but you can paraphrase it differently of course. Hope this helps.

(I had problem using quotation marks and apostrophes without the backslash showing up in each instance, so I had to use <> instead.)