Question A:
"The Achaemenid empire of Persia reached the Indus valley in the fifth century B.C., bringing the Aramaic script with it, from which was derived both northern and southern Indian alphabets"
o with it the Aramaic script, from which derives both northern and
o with it the Aramaic script, from which derive both the northern and the (**CORRECT choice**).
Why is "the" necessary before alphabets and how do I know its derives and not derive? (edit: original had typos)

"The" necessary before alphabets - The presence of "the" pays more emphasis on the Indian alphabets. I have not seen any question in the Official Guide that tests the rule pertaining to the usage of "the" (one of the three articles), so you do not need to worry about this.

The main discerning error in this question is the SV-Number Agreement. The correct number is plural. You can determine that based on the analysis of subject.

Pay attention to the following part of the sentence:
From which derives both northern and southern Indian alphabets.

What is the action in this clause - derives

Who is doing this action - i.e. what derives?

The answer is - northern and southern Indian alphabets = Subject = Plural

Thus the key to determine the correct number of verb (in this case derive) is to determine the subject for the verb and then to determine the number of this subject.

For e-GMAT users, this is discussed in the following concepts:
SV-Must Exist,
SV-Must Agree
SV-Always Singular
SV-Change Number

Question B:

"In 1988, the Council on Economic Priorities began publishing Shopping for a Better World, with the simple thesis of consumers having the power to change companies by the simple expedient of refusing to buy"
o with a thesis that is a simple one: consumers have
o whose thesis was simple: consumers have (**CORRECT choice**)
I didn't think the use of "whose" was appropriate here, why am I wrong?

This question tests your knowledge of modifiers.

Note that per OG, the modifier "whose" can be used for both persons and things. Your perception that "whose" is inappropriate stems from the belief that "whose" can only be used for persons.

That being said, now lets evaluate the choice "with a thesis..."

There are two errors with this modifier.

1. Notice the verb-ing modifier - having the power to change.... Since this modifier is not separated by a comma, it modifies the preceding noun - consumers. This implies that by itself, the expression "with the thesis of consumers" should make sense. But it does not since the thesis is not of consumers. The thesis is that consumers have the power...
2. It is not clear what the modifier "with the thesis..." modifies. Does it modify the publication "Shopping for a Better World" or does it modify the "Action of Council beginning to publish".
a. Tom left the house with the purpose of going to his office - Here clearly the modifier "with the purpose..." modifies the action of Tom leaving the house.
b. Mary took a nap with the intention of refreshing herself - Here clearly the modifier "with the intention..." modifies the action of taking a nap.

For e-GMAT users, this is discussed in the following modifier concepts:
Modifiers - Types of Modifiers
Modifiers - Relative Pronoun Modifiers

Question C:
"Last week local shrimpers held a news conference to take some credit for the resurgence of the rare Kemp's ridly turtle, saying that their compliance with laws requiring that turtle-excluder devices be on shrimp nets protect adult sea turtles"
o that require turtle-excluder devices on shrimp nets protect
o requiring turtle-excluder devices on shrimp nets is protecting (**CORRECT choice**)
Why is "that" not necessary before "require"? I foolishly chose that answer even though I thought "protect" wasn't correct

In GMAT, you must select the choice that communicates the intended meaning of the sentence without any grammatical errors. So you must have a solid basis of rejecting an answer choice.

In the choice "that require...", the error is not the presence of that. The error is the use of plural number "protect" with singular subject "compliance". So you should have eliminated this choice based on this error.

There is no rule that says that "that" is not required in this construction. Its just that between the two choices, if everything else is the same, the choice that does not have "that" is more precise.

So you must follow the following order to get to the correct answer:
1. Choice should communicate intended meaning.
2. Choice should not have any grammatical error.
3. Choice should be precise.

For e-GMAT users, a standardized method with the above guidelines is used to solve all 45 questions in the application files.

Question D:
"The company announced that its profits declined much less in the second quarter than analysts had expected it to and its business will improve in the second half of the year"
o had expected and that its business would improve (**CORRECT choice***)
If the originally sentence was changed to ...."than analysts had expected them to and that it business will improve..." would it be correct also?

Had expected them to - is correct. Its just more wordy than simply stating "had expected".

Business will improve - is incorrect. In this sentence, "would" should be used since it states an possibility into the future and hence use of "conditional would" is appropriate here.

For e-GMAT users, this is discussed in the concept - Verb - Conditional.