For example if the tekufah (the beginning of the season on the Julian calendar) fell out on Sunday we'd start asking for rain in maariv on Tuesday night. It comes out there's always two days between the tekufah and asking for rain (7 times 8 is 56. So the 57 day would be a Sunday and then Wednesday would be the 60. So since the day starts at night, Tuesday night would be the beginning of the 60 day).

The Tur writes "stop asking by mussuf of Pesach." The Beis Yosef writes that "this is not precise because we don't ask for rain in Yom tov davening (so Tur should've said we stop asking on erev yom tov)." The Bach answers "(continuing to ask for rain until mussaf) makes a difference because if one accidently davened a regular shemona esrie on yom tov and remembers at the bracha of birchas hashanim (that he's davening the wrong shemona esrie the halacha is that), he should finish that bracha like it says in siman 268. (This is the scenario where the Turs ruling would make a difference because) this person should say visen tal umatar." This ruling of the Bach is questionable because even by morid hageshem we should stop saying it at maariv. We wait until mussaf to stop because we don't want to have "groups and groups" like I said in siman 114 (see first Magen Avraham there). However by asking for rain I already stated (also there) that having "groups and groups" isn't a concern. Therefore even an individual (who made a mistake and davened a weekday shemona esrie) should not mention as soon as its maariv. This (that we stop asking as soon as it's maariv) is stated explicitly in mesechtas taanis daf 4b, it says "do we ask for rain on yom tov (certainly not!)" And (another argument against Bach) I quoted before (in siman 114 from the Yirushalmi which says) "have the yom tov have dew because rain isn't good (people don't enjoy. So just like we don't say morid hageshem which is just a praise, we certainly shouldn't ask for rain). It appears to me that the same halacha should be true when the time to begin asking for rain falls out on friday night, if one davens a weekday shemona esrie he should not mention visen tal umatar since the congregation didn't start yet and this (that the individual follows what the congregation is doing) is also implied from the Tur (he quotes the Raavya who holds need 60 full days before asking for rain. So if the congregation davened early in the day before the full 60 days, an individual davening later also doesn't say it). The Darchie Moshe writes another answer to the Beis Yosefs question (on why the Tur said stop asking by mussaf). He writes that our custom is to say in mussaf on Pesach piyyutim about dew (tal) to inform everyone that we should stop asking for rain and begin davening on dew like the Sephardic custom to say morid hatal during the summer (when the Tur said "we stop asking by mussaf," he was referring to this passive method of stopping by saying it's the time for tal and not geshem):" In birchas hashanim (in shemona esrie) one should say "satiate me from your goodness" and shouldn't say "satiate me from its goodness." Because (from your goodness) is the proper version of tefillah and I haven't heard or seen someone saying "from its goodness." One should not change a bracha from the form the Rabbis made it. This is a quote from the Roshs responsa klal 4 siman 20 and the Mahrshal writes this as well.

Even though its permitted to add on to the brachas something related to it like it says in siman 119 (the halachas of exactly how you can do this. So why can't you ask for rain, if your land needs it, in birchas hashanim instead of shomea tefillah)? The Ran answers that "rain is different because it damages in the majority of the world (for the places that don't need it then. So only can mention personal requests in the other brachas if they don't damage others)." The Bach writes that he has a tradition that one should be careful not to ask for rain, at all even in shomea tefillah, at a time the Rabbis didn't say you had to. If there's a need for rain they should appease Hashem through fasts and slichos and say verses and psalms about rain. But don't say visen tal umatar (in shemona esrie). I heard that two great Rabbis told there congregation to say visen tal umatar in shomea tefillah at a time when rain was being withheld and they (the Rabbis) died that year. There death was attributed to "troubling heaven." Thats all a quote from the Bach. See siman 575 sief 9 and the Roshs responsa klal 4 siman 10 where he writes that even on shabbos where we don't fast we say the 13 attributes (of mercy) and verses about rain. See there and see siman 2.

It's implied from the Shulchan Aruch that only if rain is being withheld (are you not obligated to go back) but otherwise you would repeat shemona esrie even though rain in that place is not a curse. This ruling needs an explanation. Nevertheless during the harvest (if ask for rain) everyone repeats shemona esrie (it's bad for every place then).

If he remembers that he forgot before he begins (the next bracha) tekah bishofer, then he says it (visen tal umatar) there, like I said in siman 114 sief 6 (that if you forget something, which you'll have to repeat shemona esrie for, can say before you start the next bracha) [Tur]. See the end of this siman.

But morid hageshem isn't related to shomea tefillah since it's a praise not a plead (therefore don't have this solution to say in shomea tefillah if forgot morid hageshem).

One asks for rain first because asking for rain is a more stringent matter than saying aneinu. This is evident from the fact that you must repeat shemona esrie if forgot visen tal umatar but do not have to repeat if you forgot aneinu.