Suppose I have a linear algebraic group $G$ over a char zero field $k$.
Is there some non-trivial (i.e. not 1 nor $G$) normal closed $k$-subgroup $N$ of $G$ such that $G/N$ is connected?
Suppose I have a map $G \to G'$ of linear algebraic groups over $k$.
Is it true that $G^0 \to G'^0$, where $G^0$ is the connected component of the identity and similarly for $G'^0$?
Question 1: no, there are linear algebraic groups in characteristic zero which are simple as abstract groups: that is, they have no non-trivial normal subgroups at all. Some examples are in this MO answer.
Question 2: I assume the question is asking if $G^0$ maps into $G'^0$. If so, then yes: the image of any connected topological space is always connected.