Yes this is certainly possible, you can run arbitrary code within a github action. However, I would advise you to instead publish your docker image on a repository and simply use it via container: docker://<url_of_your_image>

That being said, if you need to follow the steps you outlined:

I'll condense your last steps into a build-my-application as they will vary depending on how you build/deploy your app.

git clone my dockerfile from a 3rd party git url

Assuming the repo you are accessing is publicly available, or the git user running the action has access. If not you may need to replace the use of curl with a more complex git command.

name: Build and Run Docker Image

on:
  push:
    branches:
      - main

jobs:
  build-and-run:
    runs-on: ubuntu-latest
    
    steps:
    - name: Download Dockerfile from external repository
      run: |
        curl -LJO https://raw.githubusercontent.com/external/repo/main/path/to/dockerfile
      working-directory: path/to/your/dockerfile/directory

    - name: Build Docker image
      run: |
        docker build -t my-docker-image .
     
    - name: Run Docker container
      run: |
        docker run -d --name my-container my-docker-image
    
    - name: Run arbitrary code inside the container
      run: |
        docker exec my-container ./build-my-application
   

Note: If have more needs other than just building the container and running a script, or you need to anchor to a particular version of docker you should specify a container for the job itself to run in i.e. a container to build your container inside.

e.g.

jobs:
  build-and-run:
    runs-on: ubuntu-latest
    container:
      image: docker:20.10.9  # Specify the Docker version you need

GitHub actions provision a virtual machine - as you noted, either Ubuntu, Windows or macOS - and run your workflow inside of that. You can then use that virtual machine to run a workflow inside a container.

Use the container specifier to run a step inside a container. Be sure to specify runs-on as the appropriate host environment for your container (ubuntu-latest for Linux containers, windows-latest for Windows containers). For example:

jobs:
  vm:
    runs-on: ubuntu-latest
    steps:
      - run: |
          echo This job does not specify a container.
          echo It runs directly on the virtual machine.
        name: Run on VM
  container:
    runs-on: ubuntu-latest
    container: node:10.16-jessie
    steps:
      - run: |
          echo This job does specify a container.
          echo It runs in the container instead of the VM.
        name: Run in container