Why my process is wrong:-How many ways are there to choose $5$ questions from three sets of $4$, with at least one from each set?

Let's compare your method with the correct solution.

Solution: As you noted, either three questions are selected from one section with one each from the other two sections or two questions each are selected from two sections and one is selected from the remaining section.

Three questions from one section and one question each from the other two sections: Select the section from which three questions are drawn, select three of the four questions from that section, and select one of the four questions from each of the other two sections. This can be done in $$\binom{3}{1}\binom{4}{3}\binom{4}{1}\binom{4}{1}$$ ways.

Two questions each from two sections and one question from the remaining section: Select the two sections from which two questions are drawn, select two of the four questions from each of those sections, and select one of the four questions from the other section. This can be done in $$\binom{3}{2}\binom{4}{2}\binom{4}{2}\binom{4}{1}$$ ways.

Total: Since the two cases are mutually exclusive and exhaustive, there are $$\binom{3}{1}\binom{4}{3}\binom{4}{1}\binom{4}{1} + \binom{3}{2}\binom{4}{2}\binom{4}{2}\binom{4}{1} = 624$$ ways to select five questions so that at least one is drawn from each of the three sections.

Why your method is wrong?

You are counting each selection in which three questions are drawn from one section and one question is drawn from each of the other sections three times, once for each way you could designate one of the three questions as the question that is drawn from that section. For example, suppose questions $A_1, A_2, A_3, B_1, C_1$ are selected. You count this selection three times.

\begin{array}{c c} \text{designated questions} & \text{additional questions}\\ \hline A_1, B_1, C_1 & A_2, A_3\\ A_2, B_1, C_1 & A_1, A_3\\ A_3, B_1, C_1 & A_2, A_3 \end{array}

You are counting each selection in which two questions each are drawn from two sections and one question is drawn from the other section four times, two times for each way you could designate one of the two question from each section from which two questions are drawn as the question that is drawn from that section. For instance, if questions $A_1, A_2, B_1, B_2, C_1$ are drawn, your method counts this selection four times.

\begin{array}{c c} \text{designated questions} & \text{additional questions}\\ \hline A_1, B_1, C_1 & A_2, B_2\\ A_1, B_2, C_1 & A_2, B_1\\ A_2, B_1, C_1 & A_1, B_2\\ A_2, B_2, C_1 & A_1, B_1 \end{array}

Notice that $$\binom{3}{1}\color{red}{\binom{3}{1}}\binom{4}{3}\binom{4}{1}\binom{4}{1} + \binom{3}{2}\color{red}{\binom{2}{1}}\binom{4}{2}\color{red}{\binom{2}{1}}\binom{4}{2}\binom{4}{1} = 2304$$

The problem with your method is that you are overcounting!

For example: if you choose question 1 from part A as one of the four from part A, but then later choose question 2 from part A as one of the two leftover, then you could end up with the same set of questions had you first chosen question 2 from A as one of the four from A, and then later question 1 from A as one of the two leftover questions. But your method counts this as two separate sets.

To select just three questions, one from each section, you are correct that there are $4\times 4\times 4$ ways to do it.

Now if you choose two more questions from the remaining $9,$ there are (using various notations) $9C2 = \binom92 = {}^9C_2 = 36$ possible ways to do that step. If you consider each resulting list of questions to be "different", then you would have $64\times 36 = 2304$ possible ways.

The thing is, as far as this question is concerned, not every sequence of questions you chose according to your method is "different" from every other. For example, choosing questions A1, B2, and C1 for the three questions (one from each section), and then A2 and A3 for the remaining two out of nine, gives the same result as choosing questions A3, B2, and C1 and then choosing A1 and A2 for the remaining two out of nine. In both cases the examinee answers A1, A2, A3, B2, and C1.