Welcome to the world of immunology! As you've noticed, it's somewhat complicated. Here are the answers to your questions:
A) All three defences can potentially be activated. The immune system doesn't "choose" immune responses- the different systems respond to anything they detect. Now, an invading pathogen will need to get past the first line of physical defences (skin, etc) to enter the body. After that, it will usually encounter the innate immune system. The innate system will detect anything that shouldn't be there, and fight back. It also alert the adaptive immune system, which then produces targeted responses specific to that pathogen. The adaptive immune system is much more powerful than the innate one, but it takes time to activate and reach full power.
B) It is not necessary to pass all these defences to cause disease. It takes time for the immune system to defeat a disease, and a pathogen can still cause disease while the immune system is fighting it. Usually (fortunately) the immune system wins, and eventually it clears the infection out. However, you can still get sick while this process is ongoing.
As to why these defences fail, and they do- pathogens have their own set of tricks to fight or confuse the immune system. Some employ techniques to hide from the immune system. Others produce toxins or chemicals that damage it. Some, like HIV, attack it directly. And some even produce chemicals that mimic the immune systems' own regulation systems, to shut it down. Some also simply just cause damage really, really fast, before the immune system can stop them. It can take several days for an immune response to reach full power, by which it's sometimes too late.
C) Like all immune responses, sometimes this method simply doesn't work. The immune system will destroy mutant cells, but it has to be able to reach these cells, recognise them as mutated, and destroy them. This usually requires the mutation to change the cell in such a way that the immune system can recognise it. Unfortunately, not all mutations do this.