We will start with a discussion that is connected to your cases.
If one forgot to daven Mincha, he davens the Shemoneh Esrei of the following Ma’ariv twice, with the second one being a makeup for the missed tefilla.[1] This is also the halacha if the missed tefilla was the full Mincha of Erev Shabbat and the tefilla of Ma’ariv is the shortened version of Shabbat. Although both of the Ma’ariv Shemoneh Esrei tefillot that he is going to daven are of Shabbat, the second still makes up for the missed Mincha.[2]
However, the matter is more complicated in a situation in which one actually davened Mincha of Rosh Chodesh but forgot Ya’aleh V’Yavo, and the next Ma’ariv is after Rosh Chodesh, which, of course, does not include Ya’aleh V’Yavo. The Tur[3] cites a machloket Rishonim whether he should recite a second Shemoneh Esrei at Ma’ariv to make up for the insufficient Mincha. The Tur and the Shulchan Aruch[4] rule that due to the doubt regarding which opinion to follow, one should say the second tefilla with the intention that, if it is not required, it should be considered a tefillat nedava (voluntary prayer).[5] Accordingly, if one omitted Ya’aleh V’Yavo on Friday afternoon and this question arises at the time of Ma’ariv of Shabbat, one does not recite a double tefilla, because tefillot nedava are not said on Shabbat.[6]
The conceptual issue behind this machloket relates to the status of a tefilla in which a crucial section was omitted. That is, when one leaves out a necessary section of Shemoneh Esrei, is it considered a valid tefilla, just that it was missing an element that needs to be made up, or is it considered an invalid tefilla? If the lacking Shemoneh Esrei is a valid tefilla, there is no point in making up the Shemoneh Esrei if the omitted sections will not be added. If, however, the lacking Shemoneh Esrei is considered invalid, it should be made up regardless.
Rav Tzvi Pesach Frank[7] connects this machloket to your first two cases, in which one omitted Ya’aleh V’Yavo and then “v’ten tal u’matar” and the reverse, without distinguishing between the cases. According to the opinion that a tefilla with an important omission is still considered a tefilla, but one repeats it in order to provide a framework in which to insert the omission, then between the two tefillot that the person recited, he said everything necessary. If, however, the lacking Shemoneh Esrei is not considered a tefilla at all, then each of the attempts was useless and a third Shemoneh Esrei is necessary.
As mentioned, this point is subject to a machloket that is not conclusively decided. Contemporary sefarim dispute whether a third Shemoneh Esrei as a tefillat nedava is proper,[8] optional,[9] or best avoided.[10] We believe that it is proper.
Rav Chaim of Brisk is widely cited[11] as distinguishing between a case of forgetting Ya’aleh V’Yavo and one of forgetting “v’ten tal u’matar.” Ya’aleh V’Yavo is an addition, whereas “v’ten tal u’matar” is part and parcel of the tefilla’s format. Thus, if v’ten tal u’matar is omitted, it is as if that tefilla was not recited at all, and a third tefilla is needed to fix the tefilla that was valid in some ways but was lacking Ya’aleh V’Yavo. The order of omission (i.e., whether he first omitted “v’ten tal u’matar” or first omitted Ya’aleh V’Yavo) should not make a difference; either way, according to Rav Chaim, the tefilla missing “v’ten tal u’matar” was ineffective and cannot validate the Ya’aleh V’Yavo recited in that tefilla.
According to Rav Chaim, if one forgot “v’ten tal u’matar” on Friday Mincha (your third case), he should recite a double tefilla of Ma’ariv even on Shabbat, in spite of the fact that voluntary tefillot are not recited then.[12] However, there are strong questions on Rav Chaim’s distinction and its application. We will mention only one: If “v’ten tal u’matar” is indeed central to its beracha, why is it halachically acceptable for one who forgot it at its usual spot to make it up later in the beracha of Shomeiah Tefilla?[13] While there likely are answers to the questions on Rav Chaim’s approach, poskim[14] do not concur with him regarding your third case. Therefore, in the event that making up a Shemoneh Esrei that was said without “v’ten tal u’matar” would not add anything new, one repeats it only during the week, when it is permissible to recite a tefillat nedava.