Rinsing the Kiddush cup and all other Kos Shel Bracha:
The cup that is used for Kos Shel Bracha [i.e. Bentching; Kiddush; Havdala; Chupa; Bris Mila] is to have Hadacha and Shetifa performed to it. Hadacha refers to washing the inside of the cup while Shetifa refers to washing the outside of the cup.
Cleaning with a cloth: An alternative to rinsing the cup with water, is to wipe it well using a cloth in a way that it appears beautiful in both its inside and outside. [Nevertheless, according to Kaballah the cup is to be cleaned specifically with water. However in a time of need, when there is no water available, one is to clean it with a napkin. The Rebbe was witnessed to clean his Kiddush cup with a napkin prior to Kiddush during the Farbregens of Shabbos.]
If the cup is already clean: If the cup is clean in both its inside and outside then it does not require any further rinsing or cleaning, so long as it does not contain any leftovers of wine [that had food dipped in it, as explained next]. [Nevertheless, according to Kaballah the cup is to receive Shetifa and Hadacha in water even if it is clean.]
If the cup contains wine: If the cup appears clean in both its inside and outside but contains leftover wine, it is not required for one to clean the cup from this wine, unless there is crumbs of food within this wine, such as wine that had bread dipped in it. [Some Poskim however conclude that it is proper to rinse the cup even if it does not contain leftovers of bread in the wine. Furthermore, according to Kaballah the cup requires Shetifa and Hadacha in water even if it is clean.]
Using the cup a second time in same meal: If one rinsed the Kiddush cup in both its inside and outside before Kiddush, [and the cup is still clean] then it does not require to be rewashed prior to being used a second time for Kos Shel Bracha, such as the second Kos by the Seder of Pesach. If one desires to reuse this cup at the end of the meal, for Kos Shel Bracha of Birchas Hamazon, then one must verify that the cup is clean from any leftover wine that had bread dipped in it. If it is not clean then it requires Shetifa and Hadacha, or to be wiped down with a cloth as explained above. [The Rebbe did not rinse the cup for the second Kos by the Seder of Pesach, as stated above. However the Rebbe was accustomed to rinse his Kiddush cup for Beirach by the Seder. The cup was visibly dirty on the outside, having contained Charoses which was dipped in the wine.]
One is to clean the cup with water, or with a cloth, in both its inside and outside prior to using it for a Kos Shel Bracha, such as Kiddush, Havdala, Bentching and the like. If the cup is clean it does not require rinsing or cleaning with a cloth. This applies even if there is leftover wine in the cup, so long as the wine does not contain food remnant. [Nevertheless, based on Kabala, one is to always rinse the cup with water prior to use, even if clean.]
If several people are making Kiddush using the same cup, one after the other, must the cup be rinsed prior to the Kiddush of each person?
No. Some however write it is proper to re-rinse the cup being it contains leftovers of another person. Others write this is not necessary unless one is particular in this matter, not to drink the leftovers of others.
 183/2; Brachos 51a; Michaber 183/1; brought also in 271/18; 296/2
 The reason: This is done in order to the cup used for the Bracha appears beautiful in both its inside and outside. [ibid]
According to Halacha [183/2] there is no need to wash the cup if it is already clean. Furthermore even if there is leftover wine it does not need to be washed unless it has pieces of bread inside. However based on Kabala one is to always wash the cup before using it for Kos Shel Bracha even when clean. [Kaf Hachaim 183/4]
 Admur ibid; M”A 183/1; M”B 183/1
 Kaf Hachaim 183/4-5 based on Zohar; Aruch Hashulchan 183/1 based on Zohar; See Darkei Chaim Veshalom 304 that he would rinse the cup prior to Birchas Hamazon. [It is however unclear if the cup was previously used during the meal.]; See also Beis Yosef 183 in name of Mordechai that “the custom of the world is to rinse the cup even if it is clean”
 Chesed Lealafim 183/2; Yifei Laleiv 183/1; Kaf Hachaim 183/5
 Admur ibid; Michaber ibid
 Kaf Hachaim 183/4 and Aruch Hashulchan 183/1 based on Zohar
 Admur ibid and 479/2; M”A 183/1; See also 473/39 which states it does not need Shetifa and Hadacha for the 2nd Kos even though it was used with wine for the first Kos.
 Elya Raba 183/1; Kaf Hachaim 183/5; See also Beis Yosef 183 in name of Mordechai that “the custom of the world is to rinse the cup even if it is clean, although if one did not drink from it during the emal then certainly this is not necessary”. Hence we see the custom is to rinse it even from plain wine.
 Kaf Hachaim 183/4 and Aruch Hashulchan 183/1 based on Zohar
 473/39; Mahrshal 88
 Heard from Rav Y.Y. Ofen; See Darkei Chaim Veshalom 304 that he would rinse the cup prior to Birchas Hamazon.
 Based on Admur 473/39 [brought above] that if the cup was already rinsed one time it does not need to be repeated, and so is the custom on the night of the Seder, not to rinse the cup by the 2nd Kos; Piskeiy Teshuvos 271/22 footnote 230 and 24 footnote 249
Are there reasons to be stringent? Despite the above ruling of the letter of the law, perhaps it is proper to be stringent to wash the cup anyways, as the Elya Raba ibid writes that it is proper to rinse the cup from its wine even if there is no bread inside; Also the Kaf Hachaim ibid writes based on Kabala that one should always rinse the cup even if it is clean, and thus perhaps the cup must be rinsed by each individual person. However see Piskeiy Teshuvos ibid footnote 249 that writes even according to the Zohar it is not required to be rinsed by the 2nd person making Kiddush, being that it was already rinsed one time. Furthermore, even according to the Elya Raba, perhaps his ruling referred specifically to a cup with leftover wine of the meal, however leftover Kiddush wine, which isn itself a Kos Shel Bracha, does not need to be washed out according to all.
 Shulchan Shlomo 271 based on Shaareiy Teshuvah 170/16 that one who gives his friend to drink from his cup is liable.
 Piskeiy Teshuvos 271 footnote 230 being that today we are no longer particular against drinking from another person cup of Kiddush, and so is the Minhag Yisrael.
 Response of Harav Y. L. Groner, Harav Eli Landa and Harav Asher Halemel Kohen; To note that by the Seder the Rebbe did not rinse his cup before the 2nd Kos even though it contained leftover wine of the 1st Kos.
 Response of Harav Y.L. Groner