The condition $ab=0 \Rightarrow ba=0$ defines what are often called reversible rings, which, for example, have the property that the set of nilpotent elements is an ideal that coincides with the prime radical.  A full matrix ring can't have this property, so you can construct counterexamples by taking any finite field $K$ and nonabelian group $G$ to which Maschke's theorem applies.  An alternative example of a non-reversible group algebra is $K[G]$ where $K$ is the field of two elements and $G$ is the dihedral group of order $8$.  Here the set of nilpotent elements does coincide with the prime radical (the ring is local artinian), but one can find elements $a,b \in K[G]$ with $ab=0$ but $ba \neq 0$.
The condition $ab=0 \Rightarrow ba=0$ defines what are often called reversible rings, which, for example, have the property that the set of nilpotent elements is an ideal that coincides with the prime radical.  A full matrix ring can't have this property, so you can construct counterexamples by taking any field $K$ and nonabelian group $G$ to which Maschke's theorem applies.  An alternative example of a non-reversible group algebra is $K[G]$ where $K$ is the field of two elements and $G$ is the dihedral group of order $8$.  Here the set of nilpotent elements does coincide with the prime radical (the ring is local artinian), but one can find elements $a,b \in K[G]$ with $ab=0$ but $ba \neq 0$.