At our shul I noticed that someone at random will usually, but not always, say the kaddish after Aleinu by Mincha even if no one is saying kaddish for a Yahrzeit. Is this correct? Can a Ger say this particular kaddish?

Mishna Berura (O.H. 55: 5) quoting Beis Yosef mentions that we recite a minimum of seven Kadeishim every day as mentioned in the verse (Tehilim 119: 164) “Seven times a day I praise You for Your righteous judgments.” He mentions that one of the four Kadeishim dedicated to Shacharis is said after Aleinu. He counts two Kaddeishim for Mincha one said before the Amida. and one after, but no Kaddish after Aleinu. Poskim mention that one should avoid reciting unnecessary Kadeishim, being in a way similar to repeating unnecessary blessings that is indeed prohibited. However, in practice it depends on the traditions of each congregation and usually a Kaddish is added after Aleinu not only when there is an Avel that recites Kaddish, but also if someone that has that tradition is present and wants to say it. A Ger can recite this or any other Kaddish as part of the Tefila. However, on question 3761 regarding if a Ger should recite Kaddish for his dear father that did not convert, but was very good and helpful to him, practically from the ‘ Chasidei Umot Haolam’, (righteous of the nations) since in Halacha, once he converted he lost the familiar connection to his father, we wrote: “Z’kan Aharon (2; 86) maintains that although it is for many a surprising tradition, it is proper in such a case to recite Kaddish for his Gentile father, although it is not an obligation. Mimamakim (3: 8) deals with reciting Kaddish for a non-Jewish woman, that saved the life of many persecuted during W.W.11. Horav Shlomo Miller’s Shlit’a opinion is that a Ger should not recite this Kaddish for his father”