Gemara

(Rav Huna): If one entered a Beis ha'Keneses in which people are praying, he begins only if he can finish before the Shali'ach Tzibur (in his repetition) will reach Modim.

(R. Yehoshua ben Levi): He begins only if he can finish before the Shali'ach Tzibur reaches Kedushah.

Rav Huna holds that an individual says Kedushah. R. Yehoshua ben Levi holds that an individual does not say Kedushah.

Rishonim

Rambam (Hilchos Tefilah 10:16): If one enters a Beis ha'Keneses and finds the Tzibur praying silently, if he can begin and finish before the Shali'ach Tzibur will reach Kedushah, he prays. If not, he waits and prays quietly word for word with the Shali'ach Tzibur, until Kedushah. He answers Kedushah with the Tzibur, and finishes the Tefilah by himself. If he began to pray before the Shali'ach Tzibur, he does not interrupt or answer Kedushah with them.

Rosh (3:18): Rav Huna permits to begin only if he will finish before Modim, for one must bow with the Tzibur at Modim. It should not look like he denies the One they bow to. The same applies if he can reach Modim or a Berachah in which we bow, when the Shali'ach Tzibur reaches Modim, since he will bow with his colleagues. The Gemara did not want to discuss this, for it is not uniform. It is unreasonable to say that for Modim d'Rabanan, one must finished his Tefilah before (the Shali'ach Tzibur reaches) Modim. When R. Tam was in the middle of his Tefilah and the Shali'ach Tzibur reached Modim, he would bow with the Tzibur only if he was in the middle of a Berachah, for we do not bow at the beginning or end of a Berachah, except for Avos and Modim (34a). However, l'Chatchilah, one should not do so, unless he must in order to be Somech Ge'ulah l'Tefilah. The same applies to Kedushah. If he will reach the Shali'ach Tzibur when he says Nakdishach, he says everything together with the Shali'ach Tzibur, word for word. This is not called a Hefsek.

R. Yonah (13a DH u'Minhago): When R. Tam was in the middle of his Tefilah and the Tzibur said Modim, he would bow with them, lest it look like he separates from the Tzibur and does not accept on himself the One whom they accept. Chachamim permit bowing only in four places, and one may not bow at the end of other Berachos. That refers to optional bowing. When there is a need, e.g. to avoid separating from the Tzibur, it is permitted. He only bows, but he does not say anything.

Poskim

Shulchan Aruch (OC 109:1): If one enters a Beis ha'Keneses and finds the Tzibur praying Shemoneh Esre, if he can begin and finish before the Shali'ach Tzibur will reach Kedushah or Kaddish, he prays.

Mishnah Berurah (1): One may go outside the Beis ha'Keneses, and then he need not wait.

Mishnah Berurah (2): This Sa'if discusses Minchah, or one who is in Birkos Keri'as Shema when the Tzibur begins Shemoneh Esre.

Magen Avraham (2): If one entered the Beis ha'Keneses after Kedushah, and if he waits until after Modim he will not be able to answer to Kaddish, and if he waits until after Kaddish he will miss Zeman Tefilah or will not be able to pray Ma'ariv with the Tzibur, he should start immediately. It is better to miss Modim than Amen Yehei Shmei Raba. He can bow in the middle of a Berachah when they bow at Modim. If he entered shortly before Kaddish, Ma'ariv b'Tzibur is more important than Amen Yehei Shmei Raba. Likewise, Kedushah is more important than Modim. Even though Amora'im agree about Modim and argue about Kedushah, we hold that an individual may not say Kedushah. Also, he can bow in the middle of a Berachah when they bow at Modim.

Magen Avraham (3): If one already heard Kedushah or Borchu, or knows that he will hear it afterwards, he need not wait. The opinion that permits an individual to say Kedushah holds that he need not wait for it! However, one must wait for Kaddish, for there is no quota (how many Kaddeshim he must hear). Perhaps this does not apply to Kaddeshim after Aleinu.

Gra (DH ha'Nichnas): The Halachah follows R. Yehoshua ben Levi even against R. Yochanan, and all the more so against Rav and all the more so against Rav Huna.

Gra (DH Oh): Kaddish is no less (important) than Kedushah.

Mishnah Berurah (5): If one did not hear Borchu and will not hear it later, and if he begins now he will not finish in time to hear it, he waits at Shirah Chadashah until he hears Borchu. The Magen Avraham and other Acharonim obligate waiting for every Kaddish before Aleinu. Derech ha'Chayim and Chayei Adam exempt if he already answered to this Kaddish.

Mishnah Berurah (4): Since one will finish in time for Kedushah, surely he will finish in time for Kaddish! We discuss Ma'ariv, or one who already heard Kedushah. The Magen Avraham taught that Kedushah is greater than (being able to bow at) Modim. I wrote below (14) in the name of Magen Giborim that it is better to say Shemoneh Esreh together with the Shali'ach Tzibur, and he will answer Kedushah and Modim. Amen Yehei Shmei Raba is greater than Tefilah b'Tzibur. We are not concerned that (he prays Minchah while) the Tzibur begins Ma'ariv, for it is still during the day.

Bi'ur Halachah (DH ha'Nichnas): If one prays at length and does not finish in time for Kedushah even if he starts at the same time as the Tzibur, perhaps he may begin, for this is proper Tefilah b'Tzibur. Also, now he must pray; the Chiyuv to answer to Kedushah is not yet upon him. Why isn't the Shulchan Aruch concerned for passing over Mitzvos?

Note: Other recent Poskim (Aruch ha'Shulchan 109:5, Salmas Chaim 215, Dinim v'Hanhagos 4:22 and Kraina d'Igreta 1:133) say that surely he starts with the Tzibur, for now he does his Mitzvah, and he need not worry lest he lose a Mitzvah later.

Rema: The Amenim after ha'Kel ha'Kadosh and Shome'a Tefilah are like Kaddish and Kedushah. If he will not finish in time, he prays only if he cannot wait because he must be Somech Ge'ulah l'Tefilah.

Magen Avraham (5): This is if he mistakenly said Ga'al Yisrael. L'Chatchilah, he should wait at Shirah Chadashah.

Kaf ha'Chayim (12): The Yerushalmi mentions these Amenim. The Mechaber did not mention them. Since the Bavli did not mention them, it holds that they are no more important than other Amenim.

Mishnah Berurah (7): It suffices if one will reach Shome'a Tefilah with the Shali'ach Tzibur. (Saying it with him shows that he agrees to it.)

Shulchan Aruch (ibid.): If he entered after Kedushah, he prays only if he can begin and finish before the Shali'ach Tzibur will reach Modim. Similarly, if he can reach Modim or a Berachah in which we bow, he prays. If he prayed in order to be Somech Ge'ulah l'Tefilah, and the Shali'ach Tzibur reached Modim, he bows only if he is in the middle of a Berachah. We do not bow at the beginning or end of a Berachah, except for Avos and Modim.

Beis Yosef (DH ha'Nichnas): The Poskim rule like R. Yehoshua ben Levi. He is not concerned for Modim at all. The Rambam connotes like this. He did not mention reaching Modim. However, Tosfos and the Rosh connote that he must finish before the Shali'ach Tzibur reaches Modim, for they explained Rav Huna and gave R. Tam's custom. The Tur connotes that Rav Huna disagrees with R. Yehoshua ben Levi (who requires finishing before Kedushah), but R. Yehoshua ben Levi agrees with Rav Huna that he must finish before Modim.

Beis Yosef (DH u'Mah she'Chasav Rabeinu): Semak and Hagahos Maimoniyos say that one must be able to finish before Modim. (Tosfos and the Rosh say that it suffices if he will reach any Berachah in which we bow.)

Beis Yosef (DH u'Mah she'Chasav Rabeinu u'Mihu): The Gemara forbids starting if he will not finish in time. It does not permit to start with intent to bow in the middle of a Berachah.

Beis Yosef (113 DH u'Mah she'Chasav v'Davka): Mahari Avuhav asked that above (Siman 109), the Tur permits starting only if he will not reach a Berachah in which we bow in time for Modim (or if needed to be Somech Ge'ulah l'Tefilah). Here he allows bowing in the middle of a Berachah! Mahari Avuhav answered that here the Tur permits only if he bows during the entire Tefilah. I answer that one may always bow in the middle. Above, we are concerned lest he not coordinate his Tefilah with when the Shali'ach Tzibur reaches Modim, and he will be at the beginning or end of a Berachah in which we may not bow then. If he intends to reach Modim or a Berachah in which we bow, even if he cannot coordinate exactly, we are not concerned for the remote chance lest he miss Modim and be in a place where he may not bow (when they say Modim). Alternatively, one may bow in the middle of a Berachah, but if all bow at the time, it looks like a Chiyuv to bow there, and one who sees this might err and bow there. Therefore, one may not start with intent to bow in the middle of a Berachah.

Taz (2): Bowing in the middle of a Berachah in a place where this seems proper to him is not a Hefsek. Bowing only because the Tzibur is bowing, but not due to the Tefilah he says, is a little like a Hefsek. Therefore, one may not rely on this l'Chatchilah.

Magen Avraham (6, citing Darchei Moshe): L'Chatchilah one may not begin with intent to bow in the middle of a Berachah. Since he is not obligated to bow there, perhaps he will forget.

Mishnah Berurah (10): Also, if one intends to bow in the middle of a Berachah, this will distract his Tefilah.

Kaf ha'Chayim (17): The Pri Chodosh forbids relying on reaching Modim with the Tzibur, for he will be distracted. I say that he will not be distracted, for he can bow also in the middle of a Berachah.

Kaf ha'Chayim (2): Similarly, one must bow with the Tzibur whenever they bow, e.g. in Aleinu.

Mishnah Berurah (8): Since he bows with the Shali'ach Tzibur, we are not concerned that he does not say Modim d'Rabanan with the Tzibur.

Kaf ha'Chayim (16): Tosfos and the Rosh say so, for the Bavli does not mention Modim d'Rabanan. Others permit beginning only if he will be able to say Modim d'Rabanan with the Tzibur.

Kaf ha'Chayim (19): Even according to the opinion that one must say Modim, one may begin Shemoneh Esreh if he will bow in Avos with the Shali'ach Tzibur, since Baruch is essentially like Modim.

Bi'ur Halachah (DH li'Kedushah): The Pri Megadim says that these two Amenim are more important than Tefilah b'Tzibur, for they override it. However, here he came in the middle, which is not full Tefilah b'Tzibur. Semichus Ge'ulah l'Tefilah in Shacharis overrides even Amen Yehei Shmei Raba, for one may not answer to Kaddish or Kedushah between Ga'al Yisrael and Shemoneh Esre. At Ma'ariv, even Tefilah b'Tzibur overrides Semichus Ge'ulah l'Tefilah (Magen Avraham 236:3).