Why is the target function minimized at (0,0) if I can get to a negative number?
Is easy to prove that $(0,0)$ is the maximum of your suggested target function, so it may be an errata of your text.

Your second question is way more interesting, the answer is no, if you change your target function to another linear function the maximum or minimum may change, suppose you change your target function to $x+y$, the minimum now is $(0,0)$, previously it was $(1,1)$.

However, as you suggested in the comments, if you only change your function by a positive scaling factor $a$, the maximum/minimum remain the same. This is easy to prove as
$$\alpha\leq \beta\iff a\alpha\leq a\beta$$
holds (I leave the details to you).

In general, there are branches of linear programming called "postoptimization" and "sensitivity analysis" in which your second question is studied in detail, for example, describing algorithms to obtain an optimal solution of the same problem for various target functions in almost the same running time of solving just one problem.