Thanks for the clear and detailed answer Max!
I’d do that, especially because, with your current parametrization, sigma is transformed and then fed to the random walk, which is then retransformed and fed to the StudentT, so it’s really hard to see what Exponential(50) implies on the outcome scale and whether it’s justified – my guess is 50 is really gigantic, especially if your data go from -2.5 to 2.5.
It’s exactly what I meant 
You could standardize your data and see whether that helps, but the difference shouldn’t be that big, as the scale is already reasonable.
Yeah it could be if the difference is that big – although I’d first check that my model makes sense before investigating my hardware; at least I’d be more sure of what the problem could be.
Not necessarily: maybe you’re using the same model, but to model a different data generating process – are you using the same data and are interested in the same phenomenon as in the tutorial?