Do I actually have to believe the dead will come back to life?
The Yigdal prayer already showed where this belief comes from. The Mishnah's list of people who lose their share in the World to Come opens with one who says the revival of the dead is not taught in the Torah. Rambam (Maimonides, the great Jewish legal scholar and thinker of the 1100s) made it the last of his thirteen principles of faith. In the same commentary he says that a person who rejects one of the principles has left the community of Israel.
Notice what these sources do not give you: a picture. Rambam says that what everyone asks about is whether the dead will rise naked or in their burial shrouds, and that hardly anyone stops to think about what the World to Come actually is.
Even the Tanach's best-known revival story is argued over. In Yechezkel (Ezekiel), God shows the prophet a valley of dry bones that come to life, and says what they stand for: the whole house of Israel, who are saying "our hope is lost." The Gemara records several opinions. Rabbi Eliezer says the dead stood up, sang, and died again. Rabbi Yehuda says it was a parable, and a true one.
Rabbi Eliezer, son of Rabbi Yosei HaGelili, says it really happened: they went up to the Land of Israel, married, and had children. Rabbi Yehuda ben Beteira then stood up and said, "I am descended from them, and these are the tefillin my father's father left me."
So the answer has layers. The Mishnah and Rambam treat the belief as basic. These sources leave a lot of room on how and when, and on whether a given story is literal. The blessing itself asks for simple trust: the note on the line "And You are trustworthy to bring the dead to life" tells of people who answer Amen before they have seen it happen.
Mishnah Sanhedrin 10:1,Rambam, commentary on the Mishnah, Sanhedrin 10:1,Yechezkel 37:11-14,Sanhedrin 92b,Abudarham, Weekday Prayers, AmidahWhat does rain have to do with bringing the dead back to life?
In winter, right in the middle of a blessing about the dead, the Amidah says "He makes the wind blow and the rain fall." That is not a modern addition. The Mishnah, about 1,800 years old, already says to mention the might of the rains in the blessing of the revival of the dead.
The Gemara asks why it goes there, and Rav Yosef answers: rain is equal to reviving the dead, so the Sages placed it in that blessing. Another teaching puts them on the same short list. Rabbi Yochanan says God holds three keys that He never handed to a messenger: the key of rain, the key of birth, and the key of reviving the dead.
Rabbi Abbahu goes further and says the day of rain is greater than the revival of the dead. His reason: reviving the dead is for the righteous, while rain falls on the righteous and the wicked alike. The Gemara points out that Rav Yosef disagrees. For Rav Yosef they are equal, not rain greater.
So the sources do not all link them the same way. For Rav Yosef, rain is equal to reviving the dead. For Rabbi Abbahu, it is greater. For Rabbi Yochanan, both are keys only God holds. Abudarham goes further: he says the revival of the dead depends on rain, and that the dead will one day come alive through dew.
Mishnah Berakhot 5:2,Berakhot 33a,Taanit 2a,Taanit 7a,Abudarham, Weekday Prayers, AmidahWhy is a blessing about God's power full of sick people and prisoners?
The blessing calls God "mighty," and then the examples are: He holds up the fallen, heals the sick, frees the prisoners, and keeps faith with those who sleep in the dust.
Abudarham reads them as small revivals. He says the list explains the words just before it, "revives the dead with great mercy." A person collapses and the doctors count him as dead, and God holds him up. A sick person reaches the gates of death, and God heals him. Prisoners are counted as dead, and God sets them free.
He ties each one to a verse: "Hashem supports all who fall" (Tehillim 145:14), "I am Hashem, your healer" (Shemot 15:26), "Hashem frees the imprisoned" (Tehillim 146:7). You have already said two of those today. Ashrei (Tehillim 145) and Halleluyah 146 come before the Shacharit Amidah.
The Adon Olam prayer told Rabbi Yochanan's teaching from the Gemara: wherever the Tanach shows God's might, His humility is right next to it. That teaching is about verses, not about this blessing. But one of its examples sounds a lot like it: "Who rides the clouds" is followed by "father of orphans and judge of widows" (Tehillim 68:5-6).
The Gemara gives more than one answer to "where is God's might?" The Atah Hu prayer tells another: that God holds Himself back and is patient with the wicked (Yoma 69b).
Abudarham, Weekday Prayers, Amidah,Megillah 31a,Yoma 69b,Tehillim 145:14,Tehillim 146:7