I've often heard "I'm driving" or "I'm driving this time" in American media almost like a cliche. But, how did the phrase become so popular?

Is "in" necessary in this sentence? What does refer to?

Excellent question. "Difference in" is usually used to refer to a change in one thing, instead of noting the dissimilarity between two things. In this case the thing is a list of things. So a "difference in how you feel", "a difference in how you think" etc. Ex: I noticed a difference in the way you play piano now. (How you play piano gas changed). Ex: There's a difference in my attitude. (My attitude has changed.) So to show that there is a change "in" is necessary.

Does the word "altitude" refer to the character's height? Is it a common expression?

"Altitude" actually refers to height or elevation, i.e. the height of something in relation to the ground or sea level. In this case, the speaker is making a joke that the length of the high heels her friend is wearing may be affecting her judgement, due to the decreased oxygen level one finds at high altitudes. Ex: The plane is flying at an altitude of 35, 000 feet above sea level. Ex: Oxygen levels decrease at high altitudes, so mountain climbers often carry oxygen tanks with them.

Is "I can't hear myself speak" a usual expression? What does it mean?

"I can't hear myself speak" a not a usual expression or a standard English phrase. Here, it means that her thoughts are so active, because she is angry, that she can't pay attention to what she is saying.

What does "thick" mean in this context? It seems it means more than thickness!

"Thick" is referring to the consistency of the mud. If a substance or liquid is described as being thick, it means that it is dense, heavy, and does not flow freely. Ex: I ordered a milkshake. It was very thick and delicious. Ex: This soup is too thick. You should add more water to it.

What does "would" mean?

"Would" is used when indicating the consequence of an imagined event or situation. Here he is using it to refer to an intention from the point of view of the past. He is telling us that, at that point in the past, it was his intention to spend the year like this. Ex: He said he would always love her. Ex: They promised that they would help.