The Minhag in Halab (Aleppo) was to adorn the Rimonim with flowers and honor the Hachamim to place them on the Sefer Torah. It is also customary in many communities to auction off the Kibud (honor) of placing the Rimonim on the Torah scroll, at least for the Yamim Noraim (High Holidays). Although some people think this is some sort of gimmick and ploy to bring in money, the truth is that this is a very significant Misva, as the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) discusses at length. In fact, it is told that in the synagogue of the famous Kabbalist Rav Salman Musafi, the honor of placing the Rimonim sold for a very high price, and congregants would fight over the privilege. Moreover, it is customary to invite a youngster to come place the Rimonim on the Torah scroll, and one reason given is that only children, whose souls are still pure and untainted by sin, are eligible to perform this lofty act. It is thus certainly worthwhile to try to fulfill the Misva of purchasing the Rimonim or purchasing the honor of placing them on the Torah.
The custom of the Baba Sali (Rav Yisrael Abuhassera, 1889-1984) was to remove the Rimonim from the Sefer Torah when the Torah was returned to the ark after the reading, and to place them in a box on the side. Our custom, however, is to remove the Rimonim from the Sefer Torah and place them to the side right when the ark is opened in preparation to remove the Torah.
If the child is not tall enough to reach the Torah, the one holding the Torah should not lower it so the child can place the Rimonim. Instead, the child should be picked up, if this is necessary.
It is customary to open the Torah after it is removed from the ark, and the congregation looks at the writing and recites several Pesukim ("Ve’zot Ha’Torah…" and so on). There is a custom to look at the letters and try to find a word that begins with the first letter of one’s name. For example, if a person is named Eliyahu, he should find a word that begins with the letter "Alef," such as "Emor." One then extends his Sisit toward the Torah and kisses them. It is also proper at that moment to bow to the Torah out of respect. The Ben Ish Hai records the custom to bow the number of times corresponding to the number of Aliyot read on that day. This means bowing three times on an ordinary Monday or Thursday, four times on Rosh Hodesh and Hol Ha’mo’ed, five times on Yom Tob, six times on Yom Kippur, and seven times on Shabbat. Everyone present in the synagogue – men, women and children – should bow out of respect for the Torah.
The time the Sefer Torah is out of the ark is considered an Et Rason – an auspicious time for our prayers to be accepted. And thus the Ben Ish Hai, in his work Keter Malchut, writes that in between the Aliyot, one should pray for whatever needs he has, such as livelihood, Shidduchim, health and so on. It is likely that this is the basis for the widespread practice of reciting a "Mi She’berach" prayer for the one who receives an Aliya, as this is an especially favorable time for prayer. Of course, while the Torah is read, one should listen attentively to the reading and not pray. However, in between Aliyot, it is proper to pray and seize the opportunity of this special Et Rason. The Ben Ish Hai writes that the Sefer Torah should be left slightly open in between Aliyot so that the sanctity of the Torah can spread throughout the synagogue during those moments. Once the Torah is removed from the ark, this is a special time for prayer until the reading is completed.
Summary: It is considered a great honor to place the Rimonim on the Sefer Torah. It is customary to open the Torah when it is taken from the ark, and one should try to find a word in the Torah that begins with the first letter of his name, and to look at that word. One should bow at that point, as well. The time when the Torah is out of the ark is considered an auspicious time for prayer, and thus it is proper in between Aliyot to pray for one’s needs.