They can test as a 7 year old if they're turning 7 by, December? anyway, whatever, you can test as a 6 year old under that circumstance.
Gyms training programs vary and it would be impossible to comment on the specifics of how any one program is run without being familiar with that gym. Should 5 year olds be doing the full regimen as if they're testing tomorrow, well, I don't think so, that certainly wouldn't be my preference. Are they ready for some of the strength training exercises, of course as long as they've been appropriately placed at that level for their ability.
Younger than 5, possibly, I don't know, I think I may be more reserved in this aspect than some. Certainly they should not be doing the bridging exercise (shoulder flex test) at that age per USAG recommendations and I also think you need to take it easy on some other things. But starting to do some basic conditioning is likely fine (again individual basis). I just don't think it's necessary and possibly inappropriate for kids to be going HOURS per week at that age, but again I may tend to be more reserved on this issue. That's just my opinion based on my experience. If training kids for competition at such young ages was a "best practice" then as a governing body we wouldn't have capped entry into the first intro competitive level at age 6 (and the first required level isn't even until age 7), in my opinion. Yes, I know there's currently lower level competitions including younger kids and while i don't object to a couple informal rec preteam meets, I'm not sure how I feel about all that either.