Is there anything we need to do (and that you can help us with) to ensure that everything works for the plugin when they make the new Website live?
Do we need to change the license URL for the plugin from https://dev.domain.org to https://www.domain.org for things to work properly? If so, can you help us with that to ensure that there is a smooth transition?
Please advise as I am not a web developer and want to sure that our new Website goes live smoothly.
It should work fine. Microthemer uses relative paths for background images, so they should work on the new domain. And so will the license. If you do notice any issues, there is an option in Preferences to regenerate all CSS, which may come in handy.