What’s best practice for your gymnasts?
How do we prevent overuse pain later in the career?
Question: Should we use light ankle weights on young gymnasts for conditioning?
Answer:
Adding very light weights to gymnasts while conditioning is an excellent idea. At my gym coaches have made sand-filled bicycle tire tubes that can be wrapped around the waist or body.

Be cautious with using wrist or ankle weights, though. The resistance is greatly increased by being at the end of the lever of the limb. The risk, of course, is possible damage to growth plates. The bike tire tubes are safer.
Question: best methods for improving shoulder / upper back flexibility?
1) Increase passive flexibility.
My favourite passive stretch is bridge with feet elevated, pushing the shoulders away from the wall. Advanced kids are allowed to progress on to walkover drills starting with at least one foot elevated.

The lower back is “rigid” and not in full extension. (The “flatter”, the better.)
2) Increase active flexibility.
Use surgical tube or skipping rope to “bounce” to full range.

Note that measuring shoulder / upper back flexibility is difficult. If you allow the gymnast to use flexibility in the lower back (e.g. bridge test) to get a higher score, they will. Yet that’s not what we want (later) on Yurchenko, Tkachev, swing on Rings, etc.
Question: stretch at the beginning or end of a practice?
This one is easy. Improving flexibility in a muscle is best done when it is fatigued. Therefore, do flexibility improvement at the end of practice.
On the other hand, if you need flexibility during the workout (and you normally do) then you need to stretch at the beginning of the workout too. Or, perhaps better, at each apparatus as you arrive. Many beam coaches have a specific complex for stretching they do on arrival at that event.
Good coaching!