1) Can we say that semi-continuous function is stronger that simple function in the sense that semi-continuous function are simple, but simple function are not semi-continuous.
2) Let $f:\mathbb R\longrightarrow \mathbb R$. Can I say that lower semi-continuous function at $x_0$ are function that are continuous at left of $x_0$ and upper semi-continuous function at $x_0$ are function that are continuous at right of $x_0$ ?
For your first question, semicontinuous functions need not be simple. Using the wikipedia article on simple function you see that simple functions are finite sums of characteristic functions. Therefore, they can only take on finitely many values, but semicontinuous functions can take on infinitely many values. Also, some simple functions are indeed semicontinuous ($1_{\mathbb{R}}$ is certainly semicontinuous). The answers to this question give examples/pointers to semicontinuous step functions.
For your second question, this does not characterize semi-continuity. For example, the function $$f(x)=\begin{cases}0&x\not=0\\1&x=0\end{cases}$$ is upper semicontinuous, but is neither left-, nor right-continuous at $0$. A similar example works for lower semicontinuous functions. See the third paragraph of examples section of the wikipedia article on semi-continuity.
