What are the cognitive effects of increasing testosterone levels in men?
I've discussed a lot of this recently on my answer to this question: Are there recent theories on physiognomy? In summary, yes.


Can't remember which of the tons I referenced mentioned well-being, but if you run a search function (Ctrl F, or ⌘F if you don't like good computers) on the text of the freely available articles I provided hyperlinks for, I guarantee you'll find it, and it was a positive relationship with testosterone.
Testosterone ($T$) might affect concentration, but I'd bet it's not a beneficial effect if so, or at least that it's moderated strongly by context. $T$ relates to spatial intelligence, but also to aggression, which may reflect some effect on impulsivity.
$T$ affects motivation, but again, I wouldn't call it a benefit, as the specific motives it affects probably aren't optimally healthy to say the least, and every action has its opportunity cost. Specifically, $T$ increases dominance and competitiveness in men, which is probably to say that it increases the need for power. It also increases sensation seeking and risk tolerance / decreases risk aversion, which is probably to say that it affects the behavioral activation and inhibition systems. Some evidence suggests testosterone interacts with competitive experiences differently for women.
For logical/math/engineering problems, $T$ is probably more unequivocally beneficial, because it relates to spatial intelligence.


$T$ is probably most studied in men; I'm not sure effects are similar for women in general. Estradiol in women may act more like testosterone in men in some ways.  I can copy content from my other answer here if desired, or expand on particular points if there is further interest. Please comment if so; I would also welcome collaborative edits.