Yes, the size also affects finetuning but much less sensitive.

Because for small-size LMs, ICL is unstable, i.e., it sometimes degrades to classifying all examples into one category. The protocol tries to ensure analyzing ICL when it works well. (For much larger-size LMs, the performance variance would be much smaller, where this step can be ignored.)

The purpose of the experiments is not to compare the performance between them. The goal is to compare the mechanisms behind them. So it doesn't affect the conclusion itself. The point is to use the same set of examples for analysis.