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Use of Chi-sqaure Which test for association between two repeat categorical variables?

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Use of Chi-sqaure test for association between two categorical variables

I wanted to test for the effect of question type on students' use of strategies. A four-item (there are four question types) written test was given to students.

The results are recorded in a contingency table like the following, where $x$ is the frequency count. For example, $x_{11}$ means there are $x_{11}$ students use "Strategy1" to solve "Type1" question

+-----------+-------+-------+-------+-------+
|           | Type1 | Type2 | Type3 | Type4 | 
+-----------+-------+-------+-------+-------+
| Strategy1 |  x_11 |  x_21 |  x_31 |  x_41 |      
| Strategy2 |  x_12 |  x_22 |  x_32 |  x_42 | 
| Strategy3 |  x_13 |  x_23 |  x_33 |  x_43 | 
| Strategy4 |  x_14 |  x_24 |  x_34 |  x_44 | 
+-----------+-------+-------+-------+-------+

Question 1: Is it valid for me to use Chi-square test for independence? Since each student (sample) contribute to 4 counts in the table, which is different from usual contingency table.

Question 2: Is there any statistical test for me to test whether a certain strategy is associated with a certain question type?