Just tested this out, and it appears that the widget picks up on the time remaining for the most recent pet entered into training, regardless of which one is set to active.

I'll pass this information along and see what they can do. :)

This is on the radar. :)

Just pushed a fix. Could you please check if this is still happening?

"• The remaining time matches the active pet no matter what I do to confuse it." - This seems to be correct behavior, yeah?

"• The remaining time does not decrease from the initial time remaining (in the widget)" - I think this is because it has a long cache on it, unfortunately. I can look into decreasing this cache but right now I think this is intentional, most sidebar widgets have a slow cache and are good for estimates more than accurate information.