When you are working with code it is almost impossible to effectively work on the same piece of code as someone else at the same time. This is why version control is preferred over collaborative software for programming languages. Markdown incorporates elements of a normal document and program code, this is most likely why there are not many options in terms of working with it collaboratively, as the code aspect of the documents mean that they are likely better handled by a version control package. If you don't want to use a version control package, then your idea of using a collaborative package for latex and including the code later is probably your best bet.
If you are willing to look into using version control, I would recommend looking into Git. It is one of the leading version control packages available and it is free to use. It does have a bit of a learning curve, but there are numerous online resources (try http://gitimmersion.com/index.html) to help you get started. While Git dosn't enable you to work on the same document as someone else at the same time, it manages the process of different people working on different versions of the same file and helps prevent the errors you mentioned in your post eg. conflict of files.
Let me know if this helps or if you have any further questions.