I need help understanding the following :
In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple)
(i) How is this equivalent to the usual definition i.e. that a semi-simple module is a direct sum of simple modules?
(ii) Does all this generalise to an object in a semi-simple category?
As to the equivalence of the two definitions both directions use Zorns lemma. Given a sub module $N$ of a direct sum of simple modules $M=\sum M_{\alpha}$, chose a maximal set $\{M_{\beta}\}$ such that $\{M_{\beta}\}\cup \{N\}$ is independent then show $$M=N\oplus \sum M_{\beta}$$
In the other direction, we need to have simple submodules of a complemented module, once we have this we can show that it is the sum of simple submodules. So to show that simple submodules exist,
Take $x \in M$ and $Z$ a maximal submodule such that $x \not \in Z$, $Z$ is has a complement, say $N$, then show that $N$ is minimal.
Unfortunately I am not competent to answer the second question.