A question about analog IC vs digital IC
The answer to your first question is yes. Many analog IC's also have transistors and even outputs that act like a switch. An example that I'm dealing with at the moment is the BISS0001, a PIR control IC which can switch one of it's pins fully high or fully low, with no inbetween (pin 2).



The second question is also likely yes. On a "purely digital" IC, you're unlikely to find any non-digital outputs, but its almost certain that it has transistors within it that are performing non-saturating functions. What those functions might be is anyone's guess, but for example the 74ls04 had some extra transistors that deal with some sort of compensation (i think?).



The way you define analog and digital hardware is somewhat blurred, because the transistors that make up the "digital" hardware are fundamentally analog devices.