good post, but not 100% technically correct regarding the reason for the nose going up as the speed increases. Technically correct (hopefully!) explanation follows - those who read on have been warned!
You really trim for a given angle of attack. This implies a given airspeed, as there is only one airspeed which for a fixed angle of attack (AoA) creates the same amount of lift as the weight of the aircraft. The aircraft, when left alone, will always return to this angle of attack. That is static stability, to speak aerodynamicese.
Now picture this: You’re flying level at a slower airspeed than trimmed. To achieve this, you must apply back pressure to the stick Holding back pressure means you will be flying at a higher AoA than trimmed. As you let go of the stick, the aircraft will go back to the trimmed AoA. This will mean lowering the nose a few degrees at the most. However, the lower AoA will decrease the lift generated. The lift is no longer equal to the weight of the aircraft. The aircraft accelerates downwards.
In other words, the flight path turns downwards.
The aircraft remains at the trimmed angle of attack, so the downwards flight path will result in the pitch attitude lowering further as the aircraft aligns itself with the relative airflow.
Then you pick up speed. The lift increases until it again equals the weight of the aircraft. Your flight path stops turning downwards.
Theoretically, you could already be in a stable descent as this point. Typically though, the dive will be a bit steeper than the power setting and trim dictates at this point. This means the speed will increase past the trimmed airspeed. Then, the lift will be greater than the weight of the aircraft which equals to an upwards acceleration. Again, your flight path turns upwards and with it, the nose comes up as the aircraft remains at the trimmed AoA.
This cycle might repeat a few times with decreasing amplitude of the oscillations in descent rate and airspeed before the aircraft finds the stable pitch angle for a constant rate of descent. If this happens, the aircraft is said to be dynamically stable in addition to statically stable.
Some aircraft can in some configurations be dynamically neutral, meaning they will continue to climb and descend alternately left to their own devices until you manually home them in on the trimmed airspeed in straight (not necessarily level) flight. The amplitude of the oscillations will not become either larger or smaller.
Yet other aircraft can become dynamically unstable. If left to their own devices, the oscillations will keep getting bigger and bigger, until they either reach an amplitude where they are stable or cumulugranitus intervenes.
In fact, most aircraft will oscillate slightly around level flight but the oscillations are typically too small to be noticed.
I thought I was doing good trying to avoid those airport hotels... and look at me now.