Is one permitted to place a kli rishon of hot cholent, or even a kli sheini of yad saledes cholent/ hot soup, on top of a plate that has on it small challah crumbs, etc. which will become cooked from the heat of the bowl?
Also, is one permitted to eat a slice of challah over a bowl of steaming hot cholent, such that some crumbs will fall in (unintentionally)? Is one permitted to eat a piece of cake over a cup of steaming hot tea?
Yes, all of the above are permitted.
All of the above are unintentional bishul. Even though in some of these cases the bishul is inevitable and hence a psik reisha, doing an action which will surely cause melacha which is generally prohibited. In these cases the melacha being done is “lo nicha lei” he has no gain, benefit or interest in them happening. As such the melacha is already downgraded to a Rabbinic prohibition.
In addition in these cases, the bishul is actually not constructive to the food, rather destructive, hence making the melacha a “kilkul”, destructive act which in it of itself is only a Rabbinic prohibition.
In these cases there is also no clear cut bishul at all, rather these are cases where we are generally stringent to treat as bishul, such as davar gush [hot solid], kli sheni [second vessel]. Also, we are dealing with bread which is already baked and there are 2 opinions as to whether it can be cooked again with liquids. While we are stringent in this matter as well, in this case considering all the above one can certainly be lenient.
See Shu”t Shevet Halevi 7:42:2, Sefer Maor Hashabbos Vol. 1, Siman 4, footnote 29 in the name of Harav Elyashiv who discuss the permissibility of pouring hot water into a cup that has drops in it from washing the cup.