Driving a manual transmission ?

In a Manual, 2nd gear is lower than third gear, so the wheels feedback back thru the transmission, clutch to the engine, forcing the to spinning faster, but that is temporary because the engine will "brake" the vehicle unless more gas is applied.

In Automatics, The shift is controlled by vacuum & internal fluid passages. The feedback from the wheels is minimal because the Automatic transmission uses a Torque Convertor, which is a Fluid coupling device. There is no solid connection off the transmission to the engine. It does not force the engine to rev up, Newer automatics do incorporate a "locking device" for cruising mode.

no it doesn't

many sports auto trans now shift down just as you do manually, older ones true, if you lift off the gas they shift 'up' to a higher gear as the engine is not working hard, so the revs reduce on ten year old cars typically.

If you floored the gas mid corner, even an old auto box would oblige by shifting down a gear or two and revs would rise.

The revolution speed of the motor is supposed to be efficient in a set range - like you pedalling a cycle;

uphill, you pick low ratios your legs fly up and doen but you creep up the hill, puff gasp,

downhill, your legs rotate slowly and gracefully but you can get 50 mph on a steepie.

Your engine is the same, auto or manual, it's just that later stuff is more sophisticated so you get better engine braking and control.

Well when you downshift in a manual you are using the engine to slow the car down, in an auto it just upshifts basically, this is why the rpm's go down, and usually the lower the better. Most of the time your car is more efficient at lower Revolutions Per Minute (RPM).

Real simple. Standards are hard connections to the engine so downshifting will increase rpm of engine regardless of pressing the accelerator.

Automatics have a pump and torque convertor that basically act as an automatic clutch. If you apply accelerator it causes increase in pressure. If you slack off accelerator then pressure drops and no connection. Its a smart clutch in lame terms.