There have been several papers published on this topic, which don't always agree.
Check out Ten Do It Better, Do They? which concludes that "that the impact of a red card depends on the minute of the expulsion and does not have an impact at all if given at the end of the first half or later" though red cards do increase the number of goals scored.
Estimating the Effect of the Red Card in Soccer says,
We estimate the effect of the red card from betting data on the FIFA World Cup 2006
and Euro 2008, showing that the scoring intensity of the penalized team drops signiﬁcantly, while the
scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a
smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an
increased number of goals when a weaker team is punished
Down to Ten: Estimating the Effect of a Red Card in Soccer finds an effect even for cards in the second half.
Being down to nine men is much rarer and probably has not been studied carefully, for lack of data if nothing else.
You also ask, "Are there ever any scenarios when playing down a man can actually improve a team's odds of winning?" I don't know of any evidence of this, but "Ten Do It Better" does note that "an old football myth suggests that...an expulsion might also be beneficial since it increases the team spirit as well as the efforts of the affected team."