I assume that if davening has concluded, one need not say Tal at all, as it is not a mandatory part of davening as it is piyutim.
Correct. In Nussach Yeshivish in Eretz Yisrael (based on the Gr"a, I assume) they do not say Tal during the repetition of the Amida, but rather (an abridged version) before the Kaddish of the silent Amida. So we see it's not part of the repetition.
But, I'm curious if the chazzan had remembered at any time during the repetition of Musaph, should he have inserted Tal there, go back to the beginning of Musaf, or just ignore it and continue Musaph?
Those Piyutim that are said after "Mechaye Hameithim" can be said at their correct place even if he forgot the earlier ones.
One would have to study the other Piyutim to see if they are relevant to other sections and if they can be moved around. Those that are Bracha-specific (i.e. explicitly refer to a specific Bracha) probably shouldn't be said anywhere else, as they may be irrelevant to another section.
Another option - Can Tal be said at any time during the day? For example, can it be included in Mincha?
Since it's simply a prayer for rain, it can be said any time. But it cannot be included in the Amida except in the 2nd Bracha where it belongs. Since we say Piyutim during Mincha on Yom Kippur, I assume it's OK to add Tal in Mincha, if everybody forgot during Mussaf.
My question would apply to Tefillat Geshem as well. I'd imagine the rules would be the same.
I see no reason why not.
What are the above assumptions based on?
- I could not find anybody who talks about the entire congregation forgetting Tal or Geshem.
- From a strict Halachic POV there's no need for the Tal/Geshem "Piyutim" - all one needs is the announcement from the Chazzan.
- Since for (old fashioned) Ashkenazim there's nothing to announce for Tal, the fact that the Chazzan skipped Mashiv Haruach in Mussaf is the signal to stop saying it in from Mincha onwards.
- Or, some announce "Machalkel Chaim" before Mussaf as a signal to stop saying it in Mussaf already.
- Those who say "Morid HaTal" can announce that before Mussaf as a signal to stop saying it in Mussaf already.
- Same for Geshem, if Mashiv HaRuach is announced before Mussaf, one doesn't say it in Mussaf, else one stops in Mincha having heard it from the Chazzan in Mussaf.
See Shulchan Aruch 114 - סימן קיד - דין הזכרת הרוח וגשם וטל - for these rules in the original.