I don't quite understand the problem when they ask for the smallest/largest integer in the domain of a function. Am I supposed to take the limit or something?

no ... just determine the domain.
For example:

Find the smallest positive integer in the domain of f(x):

x^2 - 28x - 29 > 0 (x - 29)(x + 1) > 0 x > 29 ... x < -1 so ... what is the smallest positive integer in the domain?
and for this one:

Given

, find the largest integer in which

is decreasing.

So I found the derivative, did the sign chart and found that it was decreasing at 13.5. Is this right?

largest **integer**? wouldn't that be 13?
and for this one:

has a domain of

. Find

So for this one, I found the zero of the top of the fraction, which was 17 and 2, and I made those two the domain, so 17-2 = 15. Is this correct?

correct