Just a few points which may help to give some general guidance, YMMV.
[1] Yes, the cable "can" make a difference. Look for a good aftermarket cable from a trusted brand.
[2] When using your headphone output to run through an AUX input, generally, you want to have the volume on the source (the phone) up high, and then do the controlling with your AUX preamp controls. (Remember to start with your AUX device at LOW volume, once your phone is set to say 75%, then uise volume control on your AUX amp).
[3] Regarding headphones, check the "Sensitivity rating" of the headphones, it is listed in db (decibels), and the higher the number, generally, the "more sound" you get from them compared to another pair that have a lower sensitivity. (you also need to consider the impedance).