Does chinese songs have a natural word order?
This is really a big topic that can be expanded on multiple dimensions, but to the core of your question,


  Does Chinese songs have a natural word order?


Not always, if 'natural' means grammatical and can be used in daily spoken language or formal non-poem writing.


  If I learn Chinese from new songs, do I risk learning unnatural expressions and word orders?


Yes, as you do from lyrics in any language.

Lyrics can be seen as free-form poems, which are very lax in grammar, syntax and idiom. The author may deliberately break grammatical rules or create non-existing words and usages, for the purpose of rhetoric, rhyming, style or any other reason. In lyrics, a big proportion of the lines, if not the majority, are language fragments instead of complete sentences. This is especially true in the Chinese pop songs since the late 90's. Grammar aside, they can be too rhetorical or too literary to use in daily conversations. Keeping that in mind, lyrics are still a good way to get familiar with the language and memorize vocabularies.

I can add some examples to demonstrate the point but it's nothing special given you know the phenomenon well in English lyrics. They can be added upon request.