1) HALACHAH: MENTIONING HASH-M'S KINGSHIP IN "BONEH YERUSHALAYIM"
OPINIONS: The Gemara says that we mention Malchus (Hash-m's kingship) in the blessing of "ha'Tov veha'Metiv" (the fourth blessing of Birkas ha'Mazon) more than once in order to make up for the Malchus which should have been mentioned in the previous blessing, Boneh Yerushalayim. If Boneh Yerushalayim is a blessing that is "Samuch l'Chaverta," why does it need its own mention of Malchus? A blessing that is contiguous to the preceding blessing (or blessings) does not need its own mention of Malchus! The Gemara answers that it requires a mention of Malchus Hash-m because it is not respectful to mention the kingship of human beings (that is, the Malchus of David ha'Melech) without mentioning the kingship of Hash-m.
Why, though, did the Chachamim institute a make-up mention of Malchus Hash-m in ha'Tov veha'Metiv, and not in Boneh Yerushalayim itself? The TALMIDEI RABEINU YONAH answer that even though Hash-m's kingship must be mentioned when man's kingship is mentioned, it is not respectful to mention Hash-m's kingship in the same breath as man's kingship. This would appear as though we are equating the kingships (see also ROSH 7:22). The Talmidei Rabeinu Yonah mention that there were old texts which did mention Hash-m's kingship in Boneh Yerushalayim. Those texts, though, were mistaken, as is evident from our Gemara which requires that Hash-m's kingship be mentioned in ha'Tov veha'Metiv.
This raises an interesting question. On Rosh Chodesh and Yom Tov, when we recite "Ya'aleh v'Yavo" in the blessing of Boneh Yerushalayim, should we say "Melech" in the phrase, "Ki Kel Melech Chanun v'Rachum Atah" -- "for You are a gracious and merciful King"?
(a) The AVUDRAHAM indeed writes that the word "Melech" is omitted from Ya'aleh v'Yavo in Birkas ha'Mazon for the above-mentioned reason. The MACHZOR HA'MEFURASH (Jerusalem, 1995) adds in the name of RAV YOSEF CHAIM ZONNENFELD zt'l that even on Rosh Hashanah, "Melech" is not recited in Ya'aleh v'Yavo of Birkas ha'Mazon.
(b) The REMA (OC 188:3) writes that although the Avudraham's reasoning for omitting "Melech" is correct, we do not find anyone who actually omits it in practice. The MAGEN AVRAHAM (OC 188:2) justifies the common practice of including "Melech" by suggesting that since Ya'aleh v'Yavo is a completely separate prayer from Boneh Yerushalayim (even though it is recited within that blessing), the inclusion of "Melech" in it is not considered a mention of Hash-m's kingship next to man's kingship, and it is permitted. Others add that it would not be proper to recite Ya'aleh v'Yavo differently in Birkas ha'Mazon than the way it is recited in Shemoneh Esreh, and therefore it is permitted to mention "Melech" in Ya'aleh v'Yavo.
(c) The widely used annual calendar in Eretz Yisrael records that Minhag Yerushalayim omits the word "Melech" in Ya'aleh v'Yavo of Birkas ha'Mazon, except on Rosh Hashanah (cited in the name of the MATEH EFRAIM).
HALACHAH: The custom today is to say "Melech" in Ya'aleh v'Yavo in Birkas ha'Mazon, like the Rema (although many Sidurim put the word "Melech" in parentheses). However, if one has a family custom not to say it, then one follows his family custom.
2) SHORTENING A "TEFILAH" THAT IS NOT A "BERACHAH"
QUESTION: On Shabbos in Birkas ha'Mazon, we mention "Retzeh" in the blessing of Boneh Yerushalayim. On Rosh Chodesh, we add "Ya'aleh v'Yavo" in the blessing of Boneh Yerushalayim.
RABEINU YONAH (29a; see Insights to Berachos 29:1) states the rule that in an "abridged" version of a blessing, only that which was a blessing in the original version is mentioned. Why, then, do we include "Retzeh" (on Shabbos) and "Ya'aleh v'Yavo" (on Rosh Chodesh) in Al ha'Michyah, when those prayers are not actual blessings in Birkas ha'Mazon?
ANSWER: RAV CHAIM SOLOVEITCHIK explains that those prayers sometimes are blessings in Birkas ha'Mazon, as the Gemara here teaches. When a person forgets to say "Retzeh" (or "Ya'aleh v'Yavo") and remembers after he completes the blessing of Boneh Yerushalayim but before he begins the next blessing, he recites a short blessing in lieu of "Retzeh" (or "Ya'aleh v'Yavo") that makes mention of Shabbos (or Rosh Chodesh). For that reason we include an abridged mention of Shabbos and Rosh Chodesh in Al ha'Michyah because they indeed are abridged blessings.
Rav Chaim explains further that this is the reason why in Al ha'Michyah we mention Shabbos and Rosh Chodesh after the words "u'Veneh Yerushalayim," while in Birkas ha'Mazon we say them before the blessing of Boneh Yerushalayim. These short references are not abridged versions of "Retzeh" and "Ya'aleh v'Yavo," but rather abridged versions of the blessings of Shabbos and Rosh Chodesh. Since those blessings are said only after Boneh Yerushalayim, their abridged versions are said only after "u'Veneh Yerushalayim" as well!