If a cold dairy item (such as a liquid or solid) falls into lukewarm chicken soup, and The bowl had recently been hot, is the bowl treif?
No, the bowl is kosher.
As a general rule, things only become non-kosher through heat. According the the Rema (YD 105:3) the heat must be (a) from a "כלי ראשון"--a utensil that itself has been on the fire; AND (b) if the utensil is no longer on the fire, the non-kosher item must have fallen in immediately after the the utensil was taken off the fire so that the scalding heat is still present. The Aruch Hashulchan (ad loc.) notes that most poskim agree with the Rema. The Maharshal (see Shach ad. loc.) is more stringent. He holds that items can become non-kosher through heat the hand would recoil from, even if the utensil was never on the fire. Within this framework there are various potentially mitigating factors, but it is unnecessary to discuss them, because your case only involves lukewarm soup, and therefore does not meet the most basic criterion for rendering utensils non-kosher.