I think, unit root tests are not applicable in your case because due to the seasonal factors it will not be possible to evaluate changes in process dynamics. But still, I would recommend spaghetti plots to detect potential data flaws and choose the right data transformations. Please let me recommend our ePoster presenting a framework aiming to address some relevant tasks:
Poster A Visual Framework for Longitudinal and Panel Studies (with ...
In principle, you can, it is irrelevant if data are quarterly or annual, you just need to have two dimensions (industries and time). A different question is which techniques you can apply.
If you are going to do panel time analysis, you have to do unit root analysis. Otherwise, you will get misleading results. Time is too short for this analysis. If data is available, I suggest you extend the time interval.