You should stick to the first approach. Or its variation when you calculate the separate scale for each series. That's a short answer, there are nuances though.

Your scale must be constant if your model doesn't have a scale in it. For instance, take a look at Heston model in finance. Just look at the equations, ignore the context. You see how the valatility changes with time. The first equation models the returns, and the second one models the volatility (standard deviation). The model explicitly models volatility.

In this case I could see dynamic scaling working, because then you'd have other parts of your model that are modeling the scale itself. Your model doesn't have to be stochastic volatility like Heston's but it must explicitly account for the fact that volatility changes over time somehow. Otherwise, if your scale changes and you don't deal with it, I doubt you'll get a sensible result.

1. You should stick to the first approach. Or its variation when you calculate the separate scale for each series. That's a short answer, there are nuances though.

Your scale must be constant if your model doesn't have a scale in it. For instance, take a look at Heston model in finance. Just look at the equations, ignore the context. You see how the valatility changes with time. The first equation models the returns, and the second one models the volatility (standard deviation). The model explicitly models volatility.

In this case I could see dynamic scaling working, because then you'd have other parts of your model that are modeling the scale itself. Your model doesn't have to be stochastic volatility like Heston's but it must explicitly account for the fact that volatility changes over time somehow. Otherwise, if your scale changes and you don't deal with it, I doubt you'll get a sensible result.

2. On the second question of whether to use the training set to scale. Ideally it shouldn't even matter, because your scale is constant, right? So, if it's constant then your scale shouldn't change if you calculated it on subsamples. In reality it will move a little bit, and if the change causes issues it means that your training set is different from the test set. The macro characteristics such as means and standard deviation should not change too much between subsamples. If this happens it's a sampling or data size problem. Also, if the change in scale is small, but the model breaks down it means that the model is not robust to small disturbances, a problem in and of itself, in my opinion.