On Tuesday this leads to Ashrei, on Monday to the Torah. What is it part of?
Look at what comes next. On Monday and Thursday the Torah comes out. On a day with no Torah reading you go straight to Ashrei. Siddur Sefard has a note that says so: on days when the Torah is not read, continue with Ashrei. The Kaddish is the same either way.
The Tur (a law code from the 1300s) describes both paths. For the usual order it says the leader says Kaddish, and then Ashrei. For Monday and Thursday it says the leader stands, says the Kaddish, takes out the Torah, and stands at the opening of the ark to say "Gadlu."
So this Kaddish cannot be the start of whatever comes next. The Beit Yosef (Rabbi Yosef Karo's commentary on the Tur) passes on a list from the Ra'avad (a leading rabbi in Provence in the 1100s) of where a Kaddish goes. For this spot it gives a reason: the Amidah is a commandment by itself and is not attached to what comes after. The Kaddish closes the Amidah. It does not open the next thing.
The Torah gets the same treatment from the other side. The Ra'avad's list also has a Kaddish after the Torah reading, because reading the Torah with ten people is a commandment by itself. The Kitzur Shulchan Aruch (a short guide to daily Jewish law from the 1860s) says to say a Half Kaddish after the reading, and then lift the Torah. Peninei Halakhah (a modern guide to Jewish law) calls it an extra Half Kaddish on Monday and Thursday.
So on those days the Torah reading sits between two Half Kaddishes. The first closes the prayers before it. The second closes the reading. Peninei Halakhah puts the idea in one line: a Kaddish seals each stage and lifts it toward the honor of Heaven, and from there you go on to the next stage.
Siddur Sefard, note for Monday and Thursday,Tur, Orach Chaim 131 and 134,Beit Yosef, Orach Chaim 55,Shibbolei HaLeket 8,Kitzur Shulchan Aruch 23:25,Peninei Halakhah, Women's Prayer 20:4This Kaddish stops short. Why isn't the one that asks God to accept my prayer here?
There is such a Kaddish. It is called Kaddish Titkabal, after the line it adds: "May the prayers and requests of all the House of Israel be accepted before their Father in heaven." Peninei Halakhah (a modern guide to Jewish law by Rabbi Eliezer Melamed) says the leader says it after the Amidah ends.
In the morning it does not come here. A note in Siddur Sefard puts it after Uva Letzion, the prayer that begins "A redeemer will come to Zion." On a day with no Torah reading, between the Amidah and that Kaddish you have Tachanun, this Half Kaddish, Ashrei, Lamnatzeach (Psalm 20), and then Uva Letzion itself.
Why wait? The Terumat HaDeshen (Rabbi Yisrael Isserlein, a rabbi in Austria in the 1400s) reasons that Uva Letzion and the Kaddish after it count as the finishing of the Amidah. So the leader does not say the full Kaddish until after them. That full Kaddish points back to the Amidah, he says, because it is the one that says "titkabal." After verses and words of Torah, he notes, nobody says that line.
He gives this as his own reasoning, "it appears," not as a rule from the Gemara. And in that ruling he does not mention Tachanun at all. What the sources give us is where each Kaddish goes, not a full explanation for every spot.
That leaves this Half Kaddish with a smaller job. Peninei Halakhah says a Half Kaddish is used when a Kaddish is needed in the middle of the prayers but a long break is not wanted. Here it closes Tachanun, and the next step follows right away. The Half Kaddish unit has more on what "half" means.
Peninei Halakhah, Prayer 23:7 and Women's Prayer 20:4,Siddur Sefard, note after Uva Letzion,Terumat HaDeshen 15I just asked God for help. Why does the next prayer switch to God's name?
Look at what the Kaddish asks for. That God's name be made great and holy in the world. That His kingdom rule. That His rescue sprout. Nothing in it is about your own problems, and that is a real change from the pleading right before it.
None of the sources we checked says why a Kaddish, in particular, follows pleading. What they give is the job of a Kaddish. Peninei Halakhah says the Sages set one at the end of each stage of the prayers. It seals the stage and lifts it toward the highest goal, the honor of Heaven, and from there you go on to the next stage.
Now look at how Tachanun ends in Siddur Sefard. The last verse is Tehillim 79:9: "Help us, God of our rescue, for the glory of Your name. Save us and forgive our sins, for Your name's sake." The plea already ends on God's name, and the Kaddish opens with it. No source says that was planned. It is only there on the page.
One more rule is worth comparing. Rabbi Simlai, a sage of the Gemara, taught that a person should always set out God's praise first and only then pray for his own needs. He learns it from Moshe, who praised God's greatness before he asked to cross into the Land. The Amidah itself starts with praise. Tachanun is mostly asking. The Kaddish brings praise back at the end. That is a pattern to notice, not a ruling from these sources.
Peninei Halakhah, Prayer 23:7,Tehillim 79:9,Avodah Zarah 7b