Is it possible to have docker running inside of vmware?

Yes, it's entirely possible to run Docker in a Linux VM. Docker is a light virtualization solution, it doesn't virtualize hardware so you won't be affected by problems typical for nested VMs.

Port binding may be a bit tricky though, because you'll have to somehow connect your dev-env VM in VMware with Docker VM in VirtualBox.

You may also run into problems if you want to expose host's folders to a service in a Docker container. Windows' file systems are limited compared to Linux ones in terms of permission granularity and some services don't like that. (this issue is not Docker-specific)

In general, your app isn't locked to Docker in production. Dockerized services communicate with the world through network, no different than what regular services usually do. However, an app designed with Docker (or, more generally, cloud) in mind may be more resilient to failures and easier to maintain.

VMs with Windows guests may not work, because Docker on Windows uses virtualization to host another Linux VM. Nested virtualization is not officially supported.

This blog talks about exactly how to use Docker on Windows with VMWare Workstation. You can even use Windows 7.

https://stefanscherer.github.io/yes-you-can-docker-on-windows-7/

First install the Chocolatey package manager for windows (this is a one-line command)

@"%SystemRoot%\System32\WindowsPowerShell\v1.0\powershell.exe" -NoProfile -ExecutionPolicy Bypass -Command "iex ((New-Object System.Net.WebClient).DownloadString('https://chocolatey.org/install.ps1'))" && SET "PATH=%PATH%;%ALLUSERSPROFILE%\chocolatey\bin"

then install Docker to use VMWare workstation

choco install -y docker  
choco install -y docker-machine  
choco install -y docker-machine-vmwareworkstation  

Create a default vm

docker-machine --native-ssh create -d vmwareworkstation default  

Set your environment variables

docker-machine env | iex

Now you are all set to get started.

1. Yes, you can run docker on Windows. Windows on VMWare also works. We run docker on Linux servers on VMWare.
2. The smartest would be to have docker on you production machine also, but it would be possible to copy your data out from the containers. I don't recommend that though.
   You can use docker save to save your docker images to files.
   I think it's worth mentioning that it's not recommended running several applications inside the same container and it's not recommended to store data inside your containers, you should use volumes for that.