MPGs and RPMs jumping around

Most of the rest of the planet worries about "fuel consumption" (e.g. liters per 100km) instead of "fuel economy (MPG). When your idling your fuel consumption is infinity, and your MPG is zero. The BMW's MPG display in North American cars isort of a jury rig of the fuel consumption gauge used everywhere else. For some reason known only to BMW, the MPG gauge goes to infinity (like fuel consumption would do) when the car is idling, instead of accurately going to zero.

When the automatic transmission does not lhave the lock-up torque converter locked up, there is a fluid coupling between the engine and the transmission. So, the vehicle speed and engine speed do not sync together.

Your MPG and tachometer displays look normal to me, i. e. all over the place when in a stop-and-go and low-speed environment. Your MPG goes way down when accelerating, and way up when decelerating. That's normal.

Since the MPG and tach' are on one component, they probably share the same ground wire. If there is a problem (and I'm not convinced there is), a faulty ground would be a prime suspect. With a 16 year old, 225k mile car, expect electrical gremlins from time to time.

Your digital display might also display an almost instantaneous MPG. (My E46 M3 did, but it didn't have an alalog MPG gauge, replace by an engine oil temperature gauge.) The digital instantaneous MPG analog MPG displays should be close. The digital instantaneous really takes a running average, but over a short period of time. Only BMW know how long that time it, but it's long enough to keep the display a constant value long enough to be read.