What do I say with the tzibbur if I davened already?

Question

I daven mincha earlier in the afternoon but I go to shul for "mincha/maariv" so I'm in the shul when the congregation says mincha. I have learned that one says aleinu with a minyan even if one has said it already (though I can't right now find the source) but does that apply to other parts of davening?

I presume I respond to kedusha and kaddish (and say kaddish if I am in my year or have a yahrzeit) but would I also say modim, or, after the amida, avinu malkeinu (or just respond to the 9 lines said by the shaliach tzibbur)? I assume that there is no problem repeating shir hama'alot again, but is it required or at all problematic if I choose NOT to because I did already? Are there parts of the Shabbat or Yom Tov davening that I would or specifically would not say a second time (were I, as one example, to daven with a hashkama minyan and then come back to hear a bar mitzvah boy lein, and then not leave when musaf begins). Similarly, if I have davened maariv already but come back into the room when the later minyan meets, I presume I respond to borchu, but do I say sh'ma again?

Answer 1

Shulchan Arukh OC 65:2

קרא קריאת שמע ונכנס לבית הכנסת ומצא צבור שקורין קריאת שמע צריך לקרות עמהם פסוק ראשון שלא יראה כאלו אינו רוצה לקבל עול מלכות שמים עם חביריו
If one read Shema and entered a synagogue and found a community reading Shema, he needs to read with them the first verse so as not to appear as if he doesn't want to accept the yoke of heaven with his friends

The Gra there requires one to read all 3 paragraphs of Shema. The Magen Avraham there says it is proper manners to participate as well if the congregation is reading other things such as Psalm 145. The Mishna Berura there also lists Alenu as an example of such "other things". Igrot Moshe (YD 4:61:8) says you can't say Kaddish after something unless you participated in it a bit, so if you want to say Kaddish be sure to participate in that thing (though if you already said Kaddish earlier maybe sharing is more appropriate).

Shulchan Arukh OC 215:2

השומע אחד מישראל מברך אחת מכל הברכות, אף-על-פי שלא שמע כלה מתחילתה ועד סופה, אף-על-פי שאינו חיב באותה ברכה, חיב לענות אחריו אמן
One who hears a Jew bless one of the blessings, ... even if he isn't obligated in that blessing, is obligated to answer after him "Amen."

That ought to handle a bunch more of your cases.

The Rashba (Responsum 1:249) quotes a Rav Yehonatan who prohibits reciting Kedusha a second time if you already said it earlier elsewhere, but he disagrees.

Shulchan Arukh OC 128:28

כהן שנשא כפיו ואחר כך הלך לבית הכנסת אחר ומצא צבור שלא הגיעו לברכת כהנים יכול לישא את כפיו פעם אחרת
A Kohein who raised his hands and then went to another synagogue and found a community that hadn't reached the Kohein's blessing is allowed to raise his hands another time

This is true even though generally adding on to the blessing is a prohibition of Bal Tosif. The Mishna Berura notes that he isn't obligated to go up and bless again, just he is allowed to if he wants. If he isn't going to go up the Arukh Hashulchan recommends leaving the room in advance so people don't wonder about his lineage.

Most of the rules of honoring Torah scrolls (eg. standing when it is in transit) are not part of the prayer service per se and apply always. The Mishna Berura (OC 146 sk 7) quotes earlier authorities that this applies to not speaking during the Torah reading as well even if you already heard it.