We will begin by clarifying your second question, and that should help answer your first question as well.
There is a general idea that we should publicly sanctify HaShem’s Name at least seven times daily by means of Kaddish, in line with the pasuk: “Seven [times] in the day I praised You.”[2] Yet, each type of Kaddish has a specific function, as the Pri Megadim[3] spells out. In particular, he notes that the purpose of Chatzi (abbreviated) Kaddish is to separate between different parts of tefilla (e.g., P’sukei D’Zimra and Birkot Kri’at Shema; Ashrei and Shemoneh Esrei of Mincha). After kri’at haTorah, there is a Chatzi Kaddish because kri’at haTorah was instituted as a special mitzva within tefilla. This is the case whether or not there is maftir at a given laining.
The notion of reciting Kaddish between the seven regular aliyot and maftir was a post-Talmudic innovation intended to show the distinction between the main Torah reading obligation and the added aliya, which does not count towards the requirement of seven aliyot.[4] However, that is not the reason for the existence of the Kaddish; it is simply the reason that it was placed in between the two. The reason for the Kaddish is that it is appropriate at the end of the laining.[5] This explains why there is a Kaddish after kri’at haTorah even on Mondays and Thursdays, when there is no maftir.
Laining at Mincha of Shabbat is also followed by a Kaddish, although not immediately. The Kaddish before Shemoneh Esrei relates back to kri’at haTorah.[6] The reason that it is delayed slightly is that we want people to know when Shemoneh Esrei is starting.[7] We cannot have one Kaddish immediately after the laining and another one before Shemoneh Esrei, because there is not enough of a break to justify another Kaddish.[8] The same holds true at Mincha of a fast day. Although a haftara is read, there is no Kaddish before the haftara. Rather, the Kaddish for kri’at haTorah is recited before the ensuing Shemoneh Esrei.
It should now be understood why the simple choice for the one to recite the Kaddish after kri’at haTorah is the ba’al korei. As stated, the function of the Chatzi Kaddish after laining is similar to that of the other occurrences of Chatzi Kaddish. In all of them, it is the chazan who recites the Kaddish. Since the ba’al korei is the one who leads the services of kri’at haTorah, albeit with the participation of the olim, he is the natural choice.
Nevertheless, we point out that there are those who infer from a statement of the Rashbetz that this Kaddish can be said by a mourner.[9] We do not feel it is necessary to take a strong stand on the issue, as each minhag has a basis.