Just a thought on the 2nd situation – if a simulation is TOO deterministic (you have lots of points with very little noise) it’s actually difficult for NUTS to sample, because the gradients are basically zero everywhere except for a spike at the true answer. You could try increasing eps, or use tau instead of sigma to parameterize the normal (tau is the precision, 1/sigma, so it’s easier for the sampler to propose large values from the tail than teeny tiny values very close to zero that underflow).
I’ll have a think about the other results and give you a more complete answer. Right now it seems like my first hypothesis was wrong, but I don’t have anything off the cuff to suggest (except to play with adding more noise to the simulations)