Primal-dual schema in approximation algorithms
You are asking many questions. I will only answer the first.

The answer to the first question is that in linear programming we cannot enforce the variables to be integers. Therefore the optimal solution could be fractional. Consider for example the vertex cover problem for a graph $G = (V,E)$:

$$
\begin{align*}
&amp;\min \sum_{v \in V} x_v \\
s.t. \quad&amp; x_u + x_v \geq 1 \text{ for all } (u,v) \in E \\
&amp; x_v \in \{0,1\}
\end{align*}
$$

This is an integer program whose LP relaxation is

$$
\begin{align*}
&amp;\min \sum_{v \in V} x_v \\
s.t. \quad&amp; x_u + x_v \geq 1 \text{ for all } (u,v) \in E \\
&amp; 0 \leq x_v \leq 1
\end{align*}
$$

Now take the case of the triangle. The smallest vertex cover has size 2, but the optimal solution for the LP is $(1/2,1/2,1/2)$. As you can see, it isn't integral.