Suppose "B" has been replaced on the filesystem, too. Now "A" needs to
read "B" again for some reason. The question is: is it possible that
"A" could find an incompatible version of "B" and crash or malfunction
in some other way?
This is possible, but unlikely in most cases. If "B" is a code library, then the original version would usually not be closed. "A" would continue to use the original version of "B". If you run "A" after the update, the new version of "B" would be used. During the update, there is some risk that incompatible versions could be loaded. However, due to the way code libraries are loaded this should only be a problem if "A" need functionality not present in the versions of "B" that it loaded.
Good coding practice keeps the interface to functions the same. As a result it doesn't matter much which version is loaded, other than if there were bugs fixed in the newer version.
Configuration files are a slightly different matter, but are usually read during startup. In this case, "A" would not read "B" unless a reload of the configuration was changed. Again, it would be bad coding practice to change the format or meaning of the configuration file. An incompatible version of the configuration file should have a different name, so it wouldn't cause a problem.
Why nobody update their systems by rebooting with a live CD or some similar procedure?
Shutting down and rebooting from a different version would lead to a service outage. For servers, this is generally not desired. In any case, the package manager on the running system is aware of the software and versions it has installed. Live CDs have there own list of installed software, possibly with different versions. This makes it difficult to reliably upgrade the running system from the live CD.
Live CDs are sometimes used when a new release of the O/S is being installed. In this case, the a clean installation of the O/S is usually done. This can limit the amount of unused files from the previous version being retained. It can be more effort than upgrading the live system. However, if different root partitions are used, it can limit the risk of being stuck with an unbootable partially updated system.