If $G$ is a real Lie group and $\mathfrak g$ is its Lie algebra. If $\hat{ \mathfrak g}=\mathfrak g+i\mathfrak g$ is the complexifiaction of the Lie algebra and let $\hat G$ be the associated complex Lie group.
Now is $G$ a real Lie subgroup of $\hat G$? If yes, why this is not a complexification (in general) of $G$? And can we say that every Lie group is embedded as a submanifold in a simply-connected complex Lie group?
No, $G$ doesn't have to be a real subgroup of $\hat G$. Let $G$ be the universal covering of $SL(2,\mathbb{R})$. Then $\mathfrak{g}$ is $\mathfrak{sl}(2,\mathbb{R})$ and $\hat{\mathfrak{g}}=\mathfrak{sl}(2,\mathbb{C})$. Therefore, $\hat G$ is $SL(2,\mathbb{C})$. If $G$ was a subgroup of $SL(2,\mathbb{C})$, that would make it a matrix group, but it is well-known that it is not.