When is "being a linear algebraic $k$-group" preserved?

Question

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$.

What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example

1. When is a subgroup of a linear algebraic $k$-group again a linear algebraic $k$-group?
2. Are extensions of linear algebraic $k$-groups again linear algebraic $k$-groups?
3. If $A,B$ are linear algebraic $k$-groups and $A/B$ exists, is it a linear algebraic $k$-group?

Any other useful "operation" which preserves the property of "being a linear algebraic $k$-group" is much appreciated.

EDIT1: since I seem to have confused a lot of people, I should clarify that whenever I say "subgroup", "group-theoretic", etc, what I mean is "algebraic subgroup", "in the category of algebraic groups", etc.

EDIT2: Also, I should stress that what I'm interested in is whether the resulting thing (e.g. algebraic subgroup, quotient, extension, etc) is still defined over $k$, and not over some extension $K/k$.