Is z an odd integer?
(1) z is divisible by 3.
(2) The square root of z is an integer divisible by 3.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) Statement (1) tells us that z a multiple of 3. 3 and 6 are both multiples of 3, but 3 is odd and 6 is even. So Statement (1) is insufficient.

Statement (2) tells us that z must be a perfect square that is also a multiple of 3. Some possible values for z are 9 and 36. 9 is odd and 36 is even, so Statement (2) is also insufficient.

When combined, the statements still do not provide sufficient information. If z is equal to either 9 or 36, both statements are true, so we do not know whether z was odd.

Since the statements are both insufficient, even when combined, the correct answer is choice (E).
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Statement 2 states that "the square root of z is an integer divisible by 3". It's not z, that is divisible by 3, it's its square root.
In your explanation- 36 as an even possibility to discard statement 2 is not valid, for the root of 36 (4) is not divisible by 3. Therefore, option B is sufficient.

√36 = 6.
6 is divisible by 3, while 36 is even.

However, it's good that you challenge the explanations trying to fully understand every step. Once you spend more time thinking it through - you'll have a better understanding in future similar questions and topics.