How to use a push button & transistor as a kickstarter for power supply?
Your idea is sound, but your implementation is flawed.
First, you don't want to bother with &quot;wake&quot; and things like that - when the chip gets power it just starts from scratch - that's your &quot;wake&quot; signal. Your code will just be &quot;Hold it on&quot; and &quot;Make a sound&quot; followed by &quot;Let it switch off&quot;.
You only need one IO pin for that - the &quot;Hold it on&quot; signal. The &quot;catch&quot; signal in your schematic is just the chip getting power.
And then there's your schematic.  You should really use a P-channel MOSFET to do the switching of the power, not an NPN transistor. An NPN won't work, and a PNP (which would be what you would use for this if you were using a BJT) would impose a voltage drop, which you don't want.
Your circuit would in fact need to look like this:



simulate this circuit &ndash; Schematic created using CircuitLab
When idle M1 (P-channel with threshold voltage well below the battery voltage) is turned off by R1 pulling the gate up to the source voltage. Pressing the button pulls the gate below its threshold voltage allowing M1 to turn on. The chip then gets power.
M2 (N-channel with threshold voltage well below the chip voltage) is then activated by the chip by driving the GPIO HIGH to bypass the switch and connect the gate of M1 to GND.  R2 keeps M2 turned off while idle.
M2 is needed so that there is no current flow between the battery and the GPIO when the power is turned off.  Without it current can flow through R1 into the GPIO, through the ESD protection diodes inside the chip, and potentially break the chip.
Now to help clear up some of your questions:

Truth: Is is possible for a transistor as relay to power a chip on demand?

Yes, it is.

Risks: Is the power going to be stronger due to the amplifying nature of the transistor?

No. Amplification, with a BJT, is really a misnomer. It doesn't &quot;amplify&quot; a signal, it takes one small signal and uses that to proportionally control a larger signal. But that's only in the &quot;linear&quot; portion of the curve. When used in &quot;saturation&quot; like this it's more like a switch - power applied to the base turns on the flow of current from collector to emitter.

What about the chip, I've read that a chip can maintain its code for 20+ years without power supply. Is this true?

Yep. Flash memory typically has a retention time in the order of 10 to 100 years. 20 years is a reasonable ballpark figure.

Goals: Is there a better way to save power? Would it be best for the chip to just have sleep enabled?

Zero power is the best low power there is. Even in the deepest sleep mode the chip will still draw power. But if you do it right there can be little difference between the current drawn by a sleeping chip and the self-discharge of a battery as it sits idle.

Name: If this works, is there a better name than the Startled Monkey?

I like the name Startled Monkey. I think I'll use that in future myself.