(1) A simple module is semisimple. Vector spaces (over division rings) are semisimple. The ring Z is not a semisimple module over itself.

Is a simple ring semisimple?

One should beware that despite the terminology, not all simple rings are semisimple. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if R is a simple ring with a minimal left/right ideal, then R is semisimple.

Are simple modules cyclic?

Every simple module is cyclic, that is it is generated by one element. Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.