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1.

Associativity has no role to play unless the precedence of operator is same.

Answer» Associativity is only needed when the operators in an expression have the same precedence. Usually + and - have the same precedence. Consider the expression 7 - 4 + 2. The result could be either (7 - 4) + 2 = 5 or 7 - (4 + 2) = 1. The former result corresponds to the case when + and - are left-associative, the latter to when + and - are right-associative. Usually the addition, subtraction, multiplication, and division operators are left-associative, while the exponentiation, assignment and conditional operators are right-associative. To prevent cases where operands would be associated with two operators, or no operator at all, operators with the same precedence must have the same associativity.
Discussion
2.

The expression of the right hand side of operators doesn't get evaluated if the left hand side determines the outcome.

Answer» Because, if a is non-zero then b will not be evaluated in the expression (a || b)
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3.

In the expression the order of Assignment is NOT decided by Associativity of operators

Answer» The equal to = operator has Right-to-Left Associativity. So it assigns b=5 then a=b.
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4.

Associativity of an operator is either Left to Right or Right to Left.

Answer» Yes, the associativity of an operator is either Left to Right or Right to Left.
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5.

Assuming, integer is 2 byte, What will be the output of the program?

Answer» The integer value 2 is represented as 00000000 00000010 in binary system. Negative numbers are represented in 2's complement method. 1's complement of 00000000 00000010 is 11111111 11111101 (Change all 0s to 1 and 1s to 0). 2's complement of 00000000 00000010 is 11111111 11111110 (Add 1 to 1's complement to obtain the 2's complement value). Therefore, in binary we represent -2 as: 11111111 11111110. After left shifting it by 2 bits we obtain: 11111111 11111000, and it is equal to "fff8" in hexadecimal system.
Discussion
6.

Two different operators would always have different Associativity.

Answer» No, Two different operators may have same associativity. Example: Arithmetic operators like ++, -- having Right-to-Left associativity. Relational operators like >, >= also have Left-to-Right associativity.
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7.

Will the expression be disallowed by the compiler?

Answer» Because, here even though the value of p is accessed twice it is used to modify two different objects p and *p
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8.

Every operator has an Associativity

Answer» Yes, Each and every operator has an associativity. The associativity (or fixity) of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. Operators may be left-associative, right-associative or non-associative.
Discussion