"Related to" is the correct idiom, so A and B can immediately be eliminated for using "related with".
A) The purring noise of a cat, although most people think that cats purr because they are content, is actually involuntary and is not directly
related with the emotion of the cat.
B) Although most people think that cats purr because they are content, the purring noise of a cat is actually involuntary and is not directly
related with the emotion of the cat.
C) Although most people think that cats purr because they are content, the purring noise of a cat is actually involuntary and is not directly related to the emotion of the cat.
D) The purring noise of a cat,
although most people think that cats purr because they are content, is actually involuntary and is not directly related to the emotion of the cat.
Placing the "although..." fragment in the middle of the phrase "The purring noise of a cat ... is actually involuntary" is awkward.
E) Although most people think that
cats are purring because they are content, the purring noise of a cat is actually involuntary and is not directly related with the emotion of the cat.
Unnecessary and awkward to use the present tense here.
Leaves only C with no major issues.