When converting from Euler angles to quaternion and back to Euler angles, the resulting Euler angles may not be the same as the original Euler angles. But are they always equivalent? I mean, if I apply either original Euler rotation or the resulting Euler rotation to an object, will the object end up in the same orientation either way?
Yes.
Every rotation can (not neccessarily uniquely) be described using Euler angles. The same holds for quaternions. Unless you made a mistake while converting, the conversion will be such that input and output of each conversion step still describe the same rotation. Therefore by transitivity the original rotation is the same as the final one, since they are both equal to the intermediate one described by quaternions.
The only problem may come from the fact that if you are not careful, some situations might cause the algorithm of your conversion to fail, e.g. result in a division by zero for the gimbal-locked case. If you deal with these singularities correctly, then the final result will be equivalent to the initial input.