Rising Up to Heaven (or Lowering the Eyes)
ונוהגין בני ספרד ליתן עיניהם למטה בשעה שאומרים קק"ק ובני אשכנז וצרפת נותנין עיניהן למעלה ונושאין גופן כלפי מעלה וסמך למנהגם מספר היכלות ברוכים אתם ליהוה שמים ויורדי מרכבה אם תאמרו ותגידו לבני מה שאני עושה בשעה שמקדישין ואומרים קק"ק ולמדו אותם שיהיו עיניהם נשואים למרום לבית תפלתם ונושאים עצמם למעלה כי אין לי הנאה בעולם כאותה שעה שעיניהם נשואות בעיני ועיני בעיניהם.
...Sephardim have the custom to cast their eyes downward when they say "Kadosh, kadosh, kadosh."
Those from Ashkenaz (Germany) and France cast their eyes upward and raise their bodies upward (as they recite "Kadosh, kadosh, kadosh"). And they support their custom from Sefer Heikhalot:
Blessed are you to G-d, (and) Heaven and Earth, those who descend in the chariots (i.e., who, through meditation, ascend to the heavens). Tell my children what I do when they (Jews in worship) sanctify and say "Kadosh, kadosh, kadosh." Teach them that their eyes should be raised high above their houses of prayer and they should raise themselves upward, because I have no enjoyment in the world like that moment when their eyes are raised to my eyes and my eyes are raised to their eyes...
Covering or Closing Your Eyes
During the Shema, we cover our eyes to improve our focus, as we are supposed to recite those six words with the maximum amount of kavanah, intention.
(ה) נוהגין ליתן ידיהם על פניהם בקריאת פסוק ראשון כדי שלא יסתכל בדבר אחר שמונעו מלכוון:
(5) The custom is to place one's hands over their faces during the recitation of the first verse in order that one will not look at something else that will prevent him from intending [his heart].
(ה) כשידליק יברך ברוך אתה יהוה אלהינו מלך העולם אשר קדשנו במצותיו וצונו להדליק נר של שבת אחד האיש ואחד האשה גם בי"ט צריך לברך להדליק נר של י"ט ובי"ה בלא שבת יש מי שאומר שלא יברך וע"ל סי' תר"י: הגה יש מי שאומר שמברכין קודם ההדלקה ויש מי שאומר שמברך אחר ההדלקה (מרדכי סוף ב"מ) וכדי שיהא עובר לעשייתו לא יהנה ממנו עד לאחר הברכה ומשימין היד לפני הנר אחר ההדלקה ומברכין ואח"כ מסלקין היד וזה מקרי עובר לעשייה וכן המנהג (מהרי"ל):
(5) 5. When you ignite, you bless: "Blessed are you H' Elokim, King of the World, who sanctified us with your commandments and commanded us to ignite the Sabbath lamp"—both the man and the wife. Also on Holidays it is necessary to bless, to ignite the Holiday lamp. And on non-Sabbath High Holidays, there is one who says that you do not bless, See below Siman 610. REMA: There is one who says that we bless before the ignition and one who says that we bless after the ignition (Mordechai). And in order that it be "before your act" you should not benefit from it until after the blessing; therefore we put a hand in front of the candle after the ignition, and then we bless, and afterward we remove the hand—this is called "before the act" and such is the practice. (Mahari"l)
"Today, the blessing is administered in the course of the prayer services, during the [prayer leader's] Repetition of the Amidah. In Jerusalem, the Birkat Kohanim rite [duchaning] is performed every morning. On days when the Musaf service is recited, the [duchaning] is performed both during Shacharit and Musaf. In many other Israeli cities, some (mostly Sephardim) perform [duchaning] every day, while others (mostly Ashkenazim) do so only on Shabbat (or only on holidays). On public fast days, the Kohanim [duchan] during Mincha as well.
In the Diaspora, many Sephardic communities [duchan] every Shabbat, while in Ashkenazi communities, [duchaning] is only performed on major holidays during the Musaf prayer. An exception to this rule is Simchat Torah, when [duchaning] is done during the Shacharit services instead."
So, to summarize, the frequency you might experience the full duchaning experience will greatly vary, from every morning in Jerusalem or in certain other Israeli synagogues, to weekly on Shabbat, to even just on major holidays.
When the Kohanim are duchaning, it is customary to cover your head with a tallit and refrain from looking at the Kohanim while they duchan. This is because, supposedly, while this is occurring, Hashem is peeking through their fingers, and if you look at Hashem you may go blind! However, most people nowadays would say that a little glance at the Kohanim is okay, but you shouldn't watch them duchan out of respect for Hashem and the Kohanim.
Why Do We Stand for Certain Prayers?
(טז) כֵּיצַד הִיא הַכַּוָּנָה. שֶׁיְּפַנֶּה אֶת לִבּוֹ מִכָּל הַמַּחֲשָׁבוֹת וְיִרְאֶה עַצְמוֹ כְּאִלּוּ הוּא עוֹמֵד לִפְנֵי הַשְּׁכִינָה. לְפִיכָךְ צָרִיךְ לֵישֵׁב מְעַט קֹדֶם הַתְּפִלָּה כְּדֵי לְכַוֵּן אֶת לִבּוֹ וְאַחַר כָּךְ יִתְפַּלֵּל בְּנַחַת וּבְתַחֲנוּנִים וְלֹא יַעֲשֶׂה תְּפִלָּתוֹ כְּמִי שֶׁהָיָה נוֹשֵׂא מַשּׂאוֹי וּמַשְׁלִיכוֹ וְהוֹלֵךְ לוֹ. לְפִיכָךְ צָרִיךְ לֵישֵׁב מְעַט אַחַר הַתְּפִלָּה וְאַחַר כָּךְ יִפָּטֵר. חֲסִידִים הָרִאשׁוֹנִים הָיוּ שׁוֹהִין שָׁעָה אַחַת קֹדֶם תְּפִלָּה וְשָׁעָה אַחַת לְאַחַר תְּפִלָּה וּמַאֲרִיכִין בִּתְפִלָּה שָׁעָה:
(16) What is meant by [proper] intention? One should clear his mind from all thoughts and envision himself as standing before the Divine Presence. Therefore, one must sit a short while before praying in order to focus his attention and then pray in a pleasant and supplicatory fashion. One should not pray as one carrying a burden who throws it off and walks away. Therefore, one must sit a short while after praying, and then withdraw. The pious ones of the previous generations would wait an hour before praying and an hour after praying. They would [also] extend their prayers for an hour.
Another note for the Kaddish: this lovely Aramaic prose is repeated, in a few variations, multiple times in a service. While there are different customs in different congregations about when you should stand or be seated for the Kaddish, there is one that you should pay particular attention to- the Kaddish Yatom, or Mourner's Kaddish. In the Ashkenazi world, as far as I've encountered, most congregations have the custom of "mourners stand and recite the Mourner's Kaddish, non-mourners are seated and only say the response phrases." I've also seen some Reform congregations where everyone stands and says the Mourner's Kaddish together, in solidarity with the mourners.
In my opinion, it doesn't really matter that much whether you sit or stand for this, since it's a matter of custom. The important thing is to observe the minhag hamakom, the custom of the place you're in, because you don't want to offend someone by making them think you think you know better than everyone else. Also, in the case of the Mourner's Kaddish, people might come to think that you are mourning someone, and I'm a bit too superstitious to invite misfortune like that.
In every flavor of Judaism (at least, that I've encountered), we rise when the Ark is open and when a Torah scroll, or "sefer Torah" in Hebrew, is being carried, and you sit down only once the Ark doors are closed and the Torah is either laying on the table to be read or in a seated person's lap. When the Torah rests, you can rest! This is, again, in the spirit of showing the Torah and the Ark the same respect as you would a king or queen. However, the same rule does not apply to a printed copy of the Torah, which is sometimes called a "chumash."
(ב) הרואה ס"ת כשהוא מהלך חייב לעמוד לפניו ויהיו הכל עומדים עד שיעמוד זה שמוליכו ויגיענו למקומו או עד שיתכסה מעיניהם: הגה אבל לפני חומשים שלנו אין צריכים לעמוד (ב"י בשם הרמב"ם) ויש מחמירים. (רשב"א סי' קמ"ד) השומע קול הנושא ספר תורה אף על פי שאינו רואה אותו חייב לעמוד (ב"י בשם הרב רבינו מנוח):
(2) If one sees a Sefer Torah in motion, one must stand before it, and all should stand until the one carrying it stops and brings it to its place, or until it is visually obscured with a cover. Rema: But before our "Chumashim" one does not need to stand (B"Y citing Rambam), although some are stringent (Rashb"a 144). If one hears the voice of the one carrying the Sefer Torah, even if one cannot see them, one must stand (B"Y citing R' Rabbeinu Manoach).
What's That Weird Swaying That People Do?
תַּנְיָא אָמַר רַבִּי יְהוּדָה: כָּךְ הָיָה מִנְהָגוֹ שֶׁל רַבִּי עֲקִיבָא, כְּשֶׁהָיָה מִתְפַּלֵּל עִם הַצִּיבּוּר — הָיָה מְקַצֵּר וְעוֹלֶה, מִפְּנֵי טוֹרַח צִבּוּר. וּכְשֶׁהָיָה מִתְפַּלֵּל בֵּינוֹ לְבֵין עַצְמוֹ — אָדָם מַנִּיחוֹ בְּזָוִית זוֹ, וּמוֹצְאוֹ בְּזָוִית אַחֶרֶת. וְכׇל כָּךְ לָמָּה? מִפְּנֵי כְּרִיעוֹת וְהִשְׁתַּחֲוָיוֹת.
With regard to one’s intent during prayer, it was taught in a baraita that Rabbi Yehuda said: This was the custom of Rabbi Akiva, when he would pray with the congregation he would shorten his prayer and go up, due to his desire to avoid being an encumbrance on the congregation by making them wait for him to finish his prayer. But when he prayed by himself he would extend his prayers to an extent that a person would leave Rabbi Akiva alone in one corner of the study hall and later find him still praying in another corner. And why would Rabbi Akiva move about so much? Because of his bows and prostrations. Rabbi Akiva’s enthusiasm in prayer was so great, that as a result of his bows and prostrations, he would unwittingly move from one corner to the other (Rav Hai Gaon).
נִשְׁמָתְהוֹן דְּיִשְׂרָאֵל אִתְגְּזָרוּ, מִגּוֹ בּוּצִינָא קַדִּישָׁא דְּדָלִיק, דִּכְתִּיב, (משלי כ׳:כ״ז) נֵר יהוה נִשְׁמַת אָדָם. וְהַאי נֵר בְּשַׁעֲתָא דְּאִתְאַחֲד (נ"א דאתדליק) מִגּוֹ אוֹרַיְיתָא דִּלְעֵילָּא, לָא שָׁכִיךְ נְהוֹרָא עָלֵיהּ אֲפִילּוּ רִגְעָא. וְרָזָא דָּא, (תהילים פ״ג:ב׳) אֱלֹהִים אַל דֳּמִי לָךְ. כְּגַוְונָא דָּא כְּתִיב, (ישעיהו ס״ב:ו׳) הַמַּזְכִּירִים אֶת יהוה אַל דֳּמִי לָכֶם, לָא שְׁכִיכוּ לְכוֹן. נְהוֹרָא דְּשַׁרְגָּא כֵּיוָן דְּאִתְאַחֲדָא גּוֹ פְּתִילָה, הַהוּא נְהוֹרָא לָא שָׁכִיךְ לְעָלְמִין, אֶלָּא מִתְנַעְנְעָא נְהוֹרָא לְכָאן וּלְכָּאן, וְלָא מִשְׁתְּכִיךְ לְעָלְמִין.
(ג) כְּגַוְונָא דָּא, יִשְׂרָאֵל, דְּנִשְׁמָתַיְיהוּ מִגּוֹ הַהוּא נְהוֹרָא דְּשַׁרְגָּא, כֵּיוָן דְּאָמַר מִלָּה חֲדָא דְּאוֹרַיְיתָא, הָא נְהוֹרָא דָּלִיק, וְלָא יֵכְלוּן אִינּוּן לְאִשְׁתַּכְּכָא, וּמִתְנַעְנֵעָן לְכָאן וּלְכָּאן, וּלְכָל סִטְרִין. כִּנְהוֹרָא דְּשַׁרְגָּא, דְּהָא (משלי כ׳:כ״ז) נֵר יהוה נִשְׁמַת אָדָם כְּתִיב.
The souls of Israel have been hewn from the Holy Lamp, as is written, “The spirit of man is the lamp of the Lord” (Prov. 20, 27). Now once this lamp has been kindled from the supernal Torah, the light upon it never ceases for an instant, like the flame of a wick which is never still for an instant.
So when an Israelite has said one word of the Torah, a light is kindled and he cannot keep still but sways to and fro like the flame of a wick.
ובתפילת שמונה עשרה יש מתנועעים ויש שאינם מתנועעים. ותלוי לפי טבעו, דאם בהתנועע הכוונה יותר טוב אצלו – נכון להתנועע. ויש אדם שכוונתו יותר ברורה כשעומד בשווה לגמרי לא יתנועע. והכל לשם שמים.
And during the Amidah there are some who sway and some who don't and it depends on the person's nature. If by swaying, his kavanah improves, then he should sway; and a person whose kavanah is clearer when he stands perfectly still should not sway - and [either option] should be done for the sake of heaven...