One thing I can tell you is that there are invariant manifolds $$ \eqalign{x_{21} &= a x_{12} \cr y_{21} &= a y_{12} \cr x_{11} &= x_{22} = y_{11} = y_{22} = 0\cr} $$ on which the system becomes $$ \eqalign{y_{12}' &= - x_{12}\cr x_{12}' &= x_{12} + y_{12} - a x_{12}^3 \cr}$$ which is similar to the van der Pol system if $a > 0$, but is unstable if $a < 0$.

One thing I can tell you is that there are invariant manifolds $$ \eqalign{x_{21} &= a x_{12} \cr y_{21} &= a y_{12} \cr x_{11} &= x_{22} = y_{11} = y_{22} = 0\cr} $$ on which the system becomes $$ \eqalign{y_{12}' &= - x_{12}\cr x_{12}' &= x_{12} + y_{12} - a x_{12}^3 \cr}$$ which is similar to the van der Pol system if $a > 0$, but is unstable if $a < 0$.

In particular the answer to (2) is Yes.