Hi there,
I've checked your requirements and came up with a few specific solution but let me first explain the minimum structural reuqirements first. This architecture consists of 3 different layers. First we need an application on the phone that can push received sms to internet. Second one is a server on the internet that can receive and save sms pushed by the app on the phone. Then comes the software running on your pc as a standalone application or a script on localhost which will pull, or receive depending on the architecture, the sms from the server on the internet.
We can create all these 3 layers or as you've requested we can only create the software running on your localhost/desktop and use already available apps for the other 2 layers. However I'm not sure what you mean by Google phone but MightyText can not handle these 2 layers since it doesn't have an API or any other way to get the data out of it. I made a little research for this, though it may not be the best option, certain research will reveal it, I think pushbullet is up to it. It exactly fills the position for the app and server layer. Thus we only have to create the desktop software and then it can receive the sms via pushbullet.
Anyway, I suppose you want to use python for this since you've listed it on skills. For a web page running on localhost that's fine but if you want a desktop application I'd suggest to use C# instead still though it's your call.
Thanks
Ibrahim