Yes, every field is an integral domain.

Yes, the domain represents all the x values on a graph. Since these x values can be non-integral numbers, the domain can contain non-integral numbers.

The are under the curve on the domain (a,b) is equal to the integral of the function at b minus the integral of the function at a

The cotangent function has domain all real numbers except integral multiples of pi./2(90degrees).

Nope. Take the commutative ring Z4 (the set {0,1,2,3} with modular arithmetic and multiplication). 1 is the identity element. But 2 x 2 = 4 = 0 while 2 is not the 0 element. So it's not an integral domain.