Do all components obey Kirchhoff's Current Law?

You are exactly right: due to the conservation of charge, which is a direct consequence of the gauge symmetry of electrodynamics and therefore an unbreakable (according to all current knowledge) law of nature, the sum of current over all possible paths summed over all time is always exactly zero. In the case where the current doesn't go through discrete conductors, it's known as Gauss's Law.

For real life electronic components, Kirchoff's current law is exact to the accuracy that all the current flows through the devices pins. This is usually a very good approximation, since any imbalance in charge tends to get balanced due to electric attraction. Some components though, such as an electron gun, break this on purpose, and therefore from a circuit perspective explicitly break Kirchoff's law. Of course if you account for the stream of electrons coming out, the current law holds again.

Now there's a small but important caveat here: the charge only has to be conserved in the end, not at each moment of time separately. That means that if there's a component that stores net charge, the current can enter there, wait for some amount of time as a charge, and the exit only later. However, no practical component stores appreciable net charge for any appreciable amount of time. This is also true of capacitors and batteries: a capacitor stores an equal amount of positive and negative charge on its plates, whereas a battery has positively charged and negatively charged ions which flow (as electric current) to meet each other when the circuit is in operation. In both cases, the net charge is zero at all times, and so the total charge is constant, and Kirchoff's current law still holds. The same also holds for Flash memories, that is, the charge stored is balanced by a hole in the semiconductor.

However, as the The Photon points out in his answer, for components such as antennas, there may be a small but finite time delay between the current entering a component and exiting it.

Nonetheless, for all practical electronics purposes, for example a complicated IC as specifically mentioned by the OP, Kirchoff's current law holds exactly.

Kirchoff's circuit laws apply to circuits of lumped elements.

If your circuit contains distributed elements, such as transmission lines and antennas, you can't count on KCL applying absolutely.

For example, in a transient analysis current may flow into an antenna momentarily, without flowing out to any other circuit node, at least until 1/2 a cycle later. If we were to do a full electromagnetic analysis of the situation, we could presumably identify a displacement current from the antenna to the surrounding ground and other circuit elements, but usually such an analysis is too complicated to be tractable.