I only started doing squat this year. Prior to this I was working out with weights for almost 10 years (with all the usual excuses for not doing legs properly or at all). After weight training for that time I learned the importance of good form so when I started doing squats it was with just the bar and as far down as I could go. Head up, back always arched, pushing through the heels, feet about shoulder width basically facing frowards.
I don't know whether guys are just kidding when they say ATG or balls to the floor etc. Do you literally mean that your backside comes into contact with the floor before you ascend?
The furthest I get is for my thighs to start to come into contact with my calves and I stop.
As for the collateral benefits of doing squat I can say everything you've heard is true. It has improved my core strength. It has helped me to rebuild upper body size and strength after shoulder surgery, while I have been very limited in the amount of upper body work I can do. I have developed thicker hips, waist and back. Reduced body fat (ok cardio helps there). And of course, my legs are thicker and stronger.