No, as far as I'm aware, there is no "shot count" to dragons in Game of Thrones and in A Song of Ice and Fire as well. It's hard to prove a negative but it's never mentioned or made into a plot point so it is highly unlikely to be something.
I think your confusion is two fold here:
How to Train Your Dragon and Game of Thrones are not the same. The worlds, universes, lores and rules of one work are not the same in the other. There are shot counts in HTTYD but not in GoT. The two are completely separate works.
Dany didn't light the trench because she couldn't see the signal. There's not much more to it than that, she couldn't see the torch signal in the snowstorm so didn't know to light the trench. It's not that Drogon, or Rhaegal, have shot counts. The only reason fire arrows and Melisandre had to step in was because Dany couldn't see the signal to act on it.
For the part about Jon and Rhaegal though. Jon was near the Godswood to stay close to Bran, he wasn't near the front gate where the signal was given. He didn't know the trenches needed lighting at that point to be able to light them and he couldn't just randomly do it because that would trap their forces.