Can CodeBuild perform the following actions?

Yes, it can.
Sorta.

CodeBuild is, basically, a command execution engine. You tell it what to run in what order and it does that.

So.
Setup code course as Github, connect your account, choose "personal repos (or something like that)", choose the branch and add a webhook for "push".
As a "build" command (my case is dotnet, so...) set "dotnet pack" and specify the "-o" option to put the generated package someplace you know where.
Then as "post_build" (not necessary to be in "post_build" it can be in "build" but whatever) set "dotnet push" with the appropriate options (read here: https://docs.microsoft.com/en-us/nuget/quickstart/create-and-publish-a-package-using-the-dotnet-cli) as you know where you put your file you can pass it to this command.
For last two, I haven't done it, yet, but I'm thinking a Github Action and/or AWS Lambda function.