Absolutely! But, there is a 'but' that comes with this:

At a FOUNDATIONAL level, having a stable and strong pelvic musculature is hugely valuable.

BUT, and this is where a lot of people get it wrong: they 1. don't APPLY the skill into the dynamic environment, and 2. they don't APPLY it in the more stimulating gym/loaded environment. 

SUMMARY: a strong and stable core is crucial for movement efficiency, BUT APPLY and LOAD UP to really take full advantage of it. 

YES!

....and here comes ANOTHER 'but': 

The evidence around core stability training to help back pain is ambiguous, because pain is incredibly multifactorial.

SO, if you have back pain, increasing your pelvic stability and strength will help, BUT you must try and apply the pelvic tilting to unload the lower back and improve the glute/hip flexor balance.  

AND, with confidence, start building more strength around the lower back and pelvis to increase the capacity of the tissue (thing squats, deadlifts etc WITH good core control).

YES!

But not too young.  I only treat/educate kids from about 12 onwards, and even then it's tricky. 

For the growing adolescent athlete, there is a huge amount of value in building their foundational strength*. Use this programme to help them bed in good practice and technique. And apply to as many different sports/play situations as possible. 

Oh, and make sure they HAVE FUN!

*Note: when going through puberty it is very likely that your child's movement patterns will go all 'new born giraffe'. THis is normal, and a process most go through, so don't panic and think more exercises are going to fix this, 'cos they won't. 

Nope. 

Since core stability training is a foundational-level of strength, it's more about understanding the how and application - vs faffing with all sorts of equipment.

I absolutely use bands and swiss balls, but only as a tool to increase stimulus and difficulty, that the athlete can take into the higher level movement patterns.