Heat Exchangers

Transferring heat to the core is mostly for breeders.
However heat exchangers with core ratings also transfer heat from the reactor to themselves, and then that heat is able to be interacted with by other components. It can be pulled away by other exchangers, or by the exchanger itself if it has side transfer ability.

Best thing to do is read the sticky describing the new reactors, then read the guide describing all the new components.

You imply you have working knowledge of real nuclear piles? Am I correct on this? Anyways, next question!

For both breeders and non-breeders, the uranium can dump its heat into one of two places - an adjacent, heat-holding component, or into the core. Most designs have the uranium dumping their heat into the core (in part because the uranium might be surrounded by other uranium, and in part because for the larger reactors, any component directly receiving all the heat would quickly melt). Thus, you need all those components with core transfer to extract the heat back out of the core, so you can spread it to your cooling components.

(You can also use the core exchange to shuffle heat around the reactor. This mostly shows up with Overclocked Vents, where you can potentially wind up pulling more heat out of the core than you can cool locally, so you put some of it back into the core with Exchangers to be re-extracted by components in a different area of the reactor.)