I felt rather confused when I was taking a course on linear algebraic groups, because I found that I could not explain explicitly some subtle differences between properties of algebraic groups and properties the groups of their rational points. (I wasn't sure whether the three famous text books (Borel, Humphreys, Springer) dealt with such questions or not (maybe implicitly?), because so far I didn't find expected solution in them.)
My questions are as follows:
(1) Let $G$ be a linear algebraic group (LAG) over an algebraically closed field $k$. Is there any characterisation (equivalent conditions) for $G$ being reductive or semisimple, by dealing with similar properties of the group of $k$-points $G(k)$?
For example, if for $G(k)$, as an abstract group, the only solvable unipotent normal subgroup (now we can't talk about closedness and connectedness) is $\{1\}$, can we deduce that $G$ is reductive over $k$?
(2) Let $G$ be a reductive group over an algebraically closed field $k$. Suppose $K/k$ is an extension and $K$ is also algebraically closed. Recall the Weyl group is constructed by first taking a maximal torus $T$ of $G$ (since $k$ is algebraically closed, $T$ is split), then compute $W(k):=(N_G(T)/Z_G(T))(k)$. (Am I right here? or is there anything imprecise?)
Then is it obvious that the $W(k)$ is isomorphic to $W(K)$? (Similarly, is it obvious that $G/k$ is reductive (resp. semisimple) iff $G/K$ (as a $K$-group) is reductive (resp. semisimple)?)
(3) Suppose $G$ is a reductive group defined over a subfield $F$ of $k$ ($k$ is algebraically closed), $G_{ss}:=[G,G]$ the derived semisimple subgroup and $Z(G)$ the center of $G$, then we know that $Z(G)(k)\times G_{ss}(k)\rightarrow G(k)$ (usual group multiplication) is surjective. But my professor reminded me that it's an important fact that on the level of $F$-rational points, $Z(G)(F)\times G_{ss}(F)\rightarrow G(F)$ needn't be surjective. Is there any explicit handy example for this phenomenon?
I tried to study the three famous texts, but so far I didn't find what I need in these texts (since I'm a beginner, I guess they might be discussed in a rather implicit way but I didn't realize that). Can any expert give me some instructions on these questions? Thanks a lot in advance!