The phrase "oozing with four kinds of melty cheese" is intended to modify "[the writer's] wife's homemade bacon mac & cheese."; therefore, it should be placed at the end of the sentence to be as close as possible to the noun phrase it is describing. The participle "oozing" could still be easily attributed to either the subject "I" or the "mac and cheese," but changing the phrase "oozing with four kinds of melty cheese" to "which oozed with four kinds of melty cheese" removes the possibility of the phrase referring to the subject "I," making the sentence less ambiguous.

The original text contains a misplaced modifier. When we have a modifying phrase at the beginning of a sentence, such as "Invented in China over three centuries ago," we know that the noun that appears right after that phrase must be the thing that the modifying phrase describes. In this case, this means the thing that was invented in China over three centuries ago. 

All of the answer choices except "Invented in China over three centuries ago, the compass has served as a navigational tool for explorers and trade merchants alike" make is sound as though explorers and trade merchants were invented in China over three centuries ago, which is not correct.

When a sentence begins with an introductory modifying clause, such as "A popular tourist destination," the noun right after that clause must be the entity that the clause describes—in this case, Central Park. Otherwise, an error known as a "dangling modifier" is created and the sentence becomes confusing and ambiguous.

Only the answer choice "Central Park was the subject of many of Edward Hooper's paintings" places the subject "Central Park" immediately after the modifying introductory clause and avoids confusion and ambiguity.

All of the other answer choices make it sound as though Edward Hooper is a popular tourist destination, which is somewhat silly and definitely incorrect.

The question is an example of a dangling modifier. The dependent clause must modify the subject of the independent clause. In this case, the subject that it modifies (ostensibly, a hiker of some sort) is absent from the sentence. The correct version includes a subject that agrees with the modifying dependent clause.

With the sentence phrased the way it is, the thing that is "caring for a pet" appears to be "responsibility." This dangling modifier needs to be changed to make it clear that "children" are "caring for a pet." The correct answer choice is "Responsibility is something children can learn by caring for a pet."

The modifier "Having tallied . . ." must modify "the election officials," so "the election officials" must be the subject of the sentence. Otherwise, the opening clause of the sentence would be considered a dangling modifier.

The modifier "Feared throughout the company..." modifies "the VP" as a person, not the "VP's reputation," which is the subject of the main clause. The subject that a modifier modifies must correspond. In other words, "the VP" has to be the subject of the following clause, not "the VP's reputation." When a dependent introductory clause modifies a subject that is not the subject of the main clause it creates what is called a dangling modifier.

After the comma we need the name of the person. In this sentence the dependent introductory clause (which is not underlined) is "her" which cannot describe a "toe." In order to fix this dangling modifier, the subject of the main clause needs to be changed to "the trapeze artist."

There are two major errors here: a modification error; and an idiomatic expression that must be in the correct form, "the idiom X is not nearly as well-known a Y as Z[is]."

As the phrase Although not… Charlemagne would be nonsensical as a description of the historians instead of Pepin the Short, the first clause must become an independent clause by inserting "he is." Moreover, the idiom must be corrected to "he is not nearly as well-known a monarch as … Charlemagne [is]."

This is a classic dangling modifier error. The dangling modifier should always be followed by the noun it modifies. In this case, the only answer choice which correctly resolves this issue also implements the passive voice. Although this might dissuade you, it is important to weigh your options and ask yourself what the nature of the problem is and which answer choice resolves that issue.