Is it possible to show syntax highlighting in the "Insert TeX code" boxes in LyX?
This would be very handy when one wants to input something longer using code.
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No, not at the moment.
According to the LyX bug tracker (http://www.lyx.org/trac/ticket/4343) this has been a longly requested feature and the LyX developers are still on it – with recent changes giving some hope that this might be available in a next major release.
For larger parts of ERT (evil red text = LyXism for inserted (La)TeX code) I generally employ an external editor and use Copy & Paste to move the code between LyX and my Editor. Instead of Copy & Paste one can also maintain the the TeX code in an external
.tex file inserted with File > Insert > Child Document (Include Type = Input), which is useful during development or for maintaining larger parts of code.