I found out how to color table cells/columns/rows thanks to the answers here and here.
I followed the examples in the documentation. In the preamble, I put:
\usepackage{colortbl}
\definecolor{lightgray}{gray}{0.9}
And at the beginning of the row, I put a TeX box with:
\rowcolor{lightgray}
It does work, i.e. the table is generated with the desired color, but LyX always displays a bunch of "Misplaced \noalign" errors.
In my case, the problem was that the "row" I wanted to color was a multicolumn row, so it wasn't actually a real "row" anymore, but only one big, single cell.
Thus, I needed to use \cellcolor{lightgray} instead of \rowcolor{lightgray}.
This problem also occurs when you accidentally format the ERT / TeX code, for example by making it bold like this:
Select the whole ERT (not just the contents, but the whole red box). Remove the formatting, for example by pressing Ctrl+B, so it looks like this again:
