1. The first question whose answer is B is question
A) 1   B) 2   C) 3   D) 4   E) 5

2. The only two consecutive questions with identical answers are questions
A) 6 and 7   B) 7 and 8   C) 8 and 9   D) 9 and 10   E) 10 and 11

3. The number of questions with the answer E is
A) 0   B) 1   C) 2   D) 3   E) 4

4. The number of quesions with the answer A is
A) 4   B) 5   C) 6   D) 7   E) 8

5. The answer to this question is the same as the answer to question
A) 1   B) 2   C) 3   D) 4   E) 5

6. The answer to question 17 is
A) C   B) D   C) E   D) none of the above   E) all of the above

7. Alphabetically, the answer to this question and the answer to the following question are
A) 4 apart   B) 3 apart   C) 2 apart   D) 1 apart   E) the same

8. The number of questions whose answers are vowels is
A) 4   B) 5   C) 6   D) 7   E) 8

9. The next question with the same answer as this one is question
A) 10   B) 11   C) 12   D) 13   E) 14

10. The answer to question 16 is
A) D   B) A   C) E   D) B   E) C

11. The number of questions preceeding this one with the answer B is
A) 0   B) 1   C) 2   D) 3   E) 4

12. The number of questions whose answer is a consonant is
A) an even number   B) an odd number   C) a perfect square   D) a prime   E) divisible by 5

13. The only odd-numbered problem with answer A is
A) 9   B) 11   C) 13   D) 15   E) 17

14. The number of questions with answer D is
A) 6   B) 7   C) 8   D) 9   E) 10

15. The answer to question 12 is
A) A   B) B   C) C   D) D   E) E

16. The answer to question 10 is
A) D   B) C   C) B   D) A   E) E

17. The answer to question 6 is
A) C   B) D   C) E   D) none of the above   E) all of the above

18. The number of questions with answer A equals the number of questions with answer
A) B   B) C   C) D   D) E   E) none of the above

19. The answer to this question is
A) A   B) B   C) C   D) D   E) E

20. Standardized test is to intelligence as barometer is to
A) temperature   B) wind-velocity   C) latitude   D) longitude   E) all of the above

Looking over the comments, I see that a few people have assumed that because choosing certain "possible" responses for a question would create inconsistencies, the puzzle has no solution.  In fact, those inconsistencies merely indicate that the supposedly "possible" responses should not be chosen.  There is one, and only one, consistent key-- and that's what we're supposed to find.

It is easiest to start with the questions that Require a certain response, rather than consider possibilities.  Writing "ABCDE" 20  times also might not hurt, so that you can keep track of eliminations.

One method of full solution:

(spoiler!)

First-- Questions 5, 20, 10&16, 6&17:  These are some questions that have an immediate answer.

5.  Obviously, question 5 has the same response as itself, E.

20.  Many standardized tests do reflect intelligence of one or another sort, but are not directly a measure.  Etc.  the answer is E, as a barometer is but indirectly, if at all, related to any of these.

10&16.  In order for the questions to reference eachother, you have to select 10. A)D, 17. D)A. 

6&17.  Since it is impossible for the responses to both be all correct and incorrect, you cannot select E, which eliminates C as well.  If 6 is A, then 17 is C, so 6 is not A.  If 6 is B, 17 is B, which states that 6 is D, a contradiction.  By elimination 6 is D, and 17 is B.

Second, there are yet more interactions available after you eliminate responses we have ruled out.

1.  The answer cannot be "E," because #5 is not B.  Also, the answer cannot be A or B because of the obvious contradiction.  So, either C)3 or D)4 is the answer.  Referencing question #3, that answer cannot be B because E has already been used 2 times.  So:

1.  D (#4 is the first with answer B)

4.  B (#4 has to be B)

Third, the consonant-vowel counting (and associated questions) seem promising.

7&8.  if 7A, then 8E.  if 7B, then 8E.  if 7C, then 8AorE.  if 7D, then 8CorE.  if 7E, then 8E.  So, the only possible responses for 8 are C or E.

Since there are exactly 5 A's, and at least 2 E's, out of C)6 and E)8, you have to select E.

3.  D (there are 3 E's, because there are 8 vowels and 5 A's)

8.  E (there are 8 vowels)  Also, eliminate E from every number except 5, 8 and 20.

12.  Since there are 8 vowels, there are 12 consonants, an even number.  So, immediately the answer to 12 is A.

15.  Immediately, since 12 is A, 15 is A.

13.  Immediately, since 15 is A, 13 is D, and no odds other than 15 have the answer "A."

2.  Because no more odds can be "A," 9 and 10 cannot be the same, and 10 and 11 cannot be the same.  Also, since no more responses can be "E," 7 and 8 cannot be the same, and 8 and 9 cannot be the same.  Therefore, the only remaining possibility is that 6 and 7 are the same, so the answer is A.

7.  Immediately, the answer is D.

Fourth, you now want to pick off the stragglers.

11.  Before 11, B is used once, and #9 is unanswered.  So A,D and E are impossible. 

if 11 is C, then 9 is B, which states that 11 should be B.  Therefore, 11 is not C, and 11 must be B.

9.  For various reasons A and E are impossible.  Also, B would contradict #11, so B is wrong.  C contradicts #12, so C is wrong.  Therefore, the response is D.

And, 13 is the next QUESTION WITH THE SAME ANSWER AS 9, both having "D."

The objection that "next" should mean #10 is quite absurd.  That grammar would have been:

The next question, with the same answer as this one, is question...

NOTE THE COMMAS

Anyway, in common usage, the whole expression would be interpreted as the next question having a repetition, and there is no logical demand that "next..." refers to question #10. 

As a note, in a self-referential, self-consistent scheme, you know that the goal is to infer the total state using interpretations that are within reasonable standards.  So, pointing out a possible interpretation that creates an inconsistency, but cannot be part of a complete, alternate answer, even if accurately, misses the point entirely.

14&18.  There must be 5 A's.  14 cannot be "A," because that would create another row.  Therefore, 18 is "A" to bring the total to 5 A's.

Since 18 now states that A=B, and A=5, you must select B exactly two more times (it has only been used 3).  So, Questions 14 and 19 are answered "B."

This should all be expected as the test is "self-referential."  As long as we've done nothing more than what is Required at any given step (in other words, we select or eliminate based on Proof rather than Possibility), we have the one consistent response.

And yes, I am stuck in a hotel in an unfamiliar city while the weather is too miserable for exploration, and the NBA playoffs are not enough to hold my exclusive attention.