Yes, the issue with the chronograph resetting when it was stopped, is likely a missing/broken stem bolt for hammer. This is easily fixed but sometimes find the parts of it requires taking the entire movement apart.
The more possibly troubling thing for me is that the bottom pusher did nothing - even if the minutes and seconds recorders are reset when the chronograph is stopped because of the stem bolt issue, the lower pusher should still reset the hour counter. Even if the hour counter hasn't moved, you should still feel the pusher moving the hammer in to reset it, so there will be some resistance there.
