Yes, you can do the replacement with . No, you cannot leave the limits as they are. They are values of x, because of the differential. You have to change them to values of . Or, if you want, you can convert the antiderivative back into x's before you evaluate at the limits. I generally find the first alternative less work, but you can do as you wish.
No, that's the integrand. You've successfully put the integrand into thetas. What you have to do now is find the differential . This is the same thing you have to do with any substitution (u substitution or whatever). In order to find the differential, I would recommend finding the simplest relation possible between x and theta from the triangle you drew. What do you get?