The convergence criterion is $\| y^{k+1} - y^{k} \| < \varepsilon$ where $k$ is iteration number.
Question 1.
Is it possible to use simpler criterion $$ \max_j \left(\left | y_j^{k+1} - y_j^k \right |\right) < \varepsilon $$ where $j$ is equation number? If yes then
Question 2.
Probably for particular equations the condition $$ \left | y_j^{k+1} - y_{j}^k \right |< \varepsilon $$ will be fulfilled earlier than for others. Is it possible to exclude these equations from the following iterations?
Q1. I guess yes, because norms are equivalent in finite dimension (but the $\varepsilon$ will not have the same value.
Q2. For each iteration, you have to check that for each $j$, $|y_j^{k+1}-y_j^k|<\varepsilon$. It could be that the max is obtained for one $j$ at one iteration, and for another one at another. So I don't think you can exclude some indexes $j$ in the criterion. Anyway, this criterion is computationally inexpensive so it shouldn't be a practical problem.