If you use "semisimple" as I do (to mean "$R$ is the direct sum of its simple right ideals" or "$R$ is Artinian with $rad(R)=0$") then the Artin-Wedderburn theorem proves the first statement true.
For the second question, I assume you are familiar with the proof that every vector space over a field $F$ has a basis. Once you know that is true, and $V$ has a basis $\{v_i\mid i\in I\}$, then you can map elements of $V$ to their coefficients in $\bigoplus_{i\in I} F$ to produce an isomorphism, showing that $V$ is free. If you review try this with division rings, you will find that commutativity was not necessary, and everything goes through here as well.
You also can try to work out the converse: if all right $R$ modules are free, then $R$ is a division ring.