As I understand, this is not possible to do
You understand correctly. At least, it's not doable using the methods you propose.
but below is what I really want.
-- Small form factor unit, the size of a doorbell transformer
-- inputs would be 120v ac + doorbell switch terminals, just like the transformer
-- some way to pair with an alexa, one is fine, multiple would be better, perhaps using a phone and the alexa app, most likely via bluetooth
-- pairing would transfer wifi info and alexa serial/address info then confirm correct setup and compatibility
If the doorbell and Echo are already communicating via Bluetooth, why bother with WiFi as well? Conversely, why bother with the Bluetooth connection if you're just going to switch over to WiFi (which you can't do anyway)? And what do you mean by "alexa serial/address info"?
-- when doorbell button is pressed, device sends signal to alexa via wifi (this does not appear to be possible with the existing api)
Yeah, this is the biggest problem. You cannot send unsolicited messages to your Echo via the Skills architecture. You could send audio data via a Bluetooth connection, but that connection would need to be maintained constantly.
-- alexa app allows selecting chime and volume, perhaps also a way to schedule silent mode
You'd need to build those smarts into the doorbell and just use the Echo as a an audio rendering device, though that would then make it incapable of being used for that purpose by other things, like your phone or a tablet.
-- alexa app also can receive push notifications so your phone alerts with the same chime as alexa
-- voice control via alexa to enable/disable doorbell sleep for a specific period of time (baby just went down for a nap) after which doorbell automatically becomes functional again
That's possible, but you'd probably have to host your Alexa skill on your own server in order to pass instructions along to the doorbell.
-- price in the $20-$30 range
Build it all yourself and you could do it for close to that (ignoring any monetary value of your time and effort). But that price range for an off-the-shelf product that does what you want? That's funny.
Then I could rip my doorbell and transformer off the wall and toss them.
Alexa could escalate volume and/or intensity based on number of presses of the doorbell or length of time doorbell is pressed, like force touch. Or perhaps different chimes for single press vs double-press.
I know I can buy a fancy $200 webcam doorbell but I just don't want or need that level of fancy tech on the outside of my house. I like my doorbell button, just hate my doorbell chime and the ugly transformer. It also seems totally redundant once you have an alexa, it can do everything necessary if it just had a way to receive the input from the doorbell switch.
The problem is that your Echo (or Dot, Tap, et al) is not a programmable (by you) general-purpose computing device or a smart device control hub.