I do not want to be dick but......surely most competent, well co-ordinated pilots would "Simultaneously" move both controls and the pedals for that matter. Is that not the fundamental principle we all learnt right at the beginning. You cannot move one control without having to move the two others!!
That's why we can pat our belly and rub our head (or some such) at the same time.
I do think however, that slamming the lever down at high speed will initiate, and lead to, a strong nose tuck for some types. Maybe this is what happened in the Moseby accident. Paradoxically, knowing he was short of fuel, he my have slammed the lever to the floor and just failed to prevent the nose tuck and the subsequent loss of control it leads to.
SAS - sorry just saw the end f your post. I have about 3k on 355s (of course two engines) but the fuselage is essentially the same shape. The nose definitely drops if the lever is lowered aggressively and I think this is similar to most types but most of us make a "joined up" control movement instinctively and the tendency is overcome without us seeing it through to its full manifestation.
Same as Flapback and inflow roll during take off run. How many of us, not doing basic instruction, actually recognise this anymore. It's become a subconscious set of contro movements in response to attitude.
DB