Your questions are a bit fuzzy in some points but I'll try my best to clarify all of them.
First off all your title question
How to donate a complete working project to the open source comunity?
In most jurisdictions, donating is considered as a form of gift for a specific purpose. For example you can donate books to a local charity or donate money to a non-profit organization. I think what you mean is to share your application, your code and your effort with the community, to help people with a specific problem, to gain intrerest in your project and to find fellow developers to work on your project together.
How you do that is ABSOLUTELY your choice! It's your code, so you are the copyright holder. As long as distributing your code does not violate the law, you are free to share it with anyone you like. It only gets complicated with other peoples code.
Which leads to the next question: Dependencies.
I'm accustomed to open-source dependencies, where I go to a central repository (E.g. Maven Repository) get a certain version of a certain jar and add it as a dependency to my project.
You didn't exactly ask it, but I'd like to say something about the deps as well. You say you use maven to add them to your project. This is fine as long as you comply the licence conditions of all 3rd-party code you use. This means that you have to look at the libraries you use and check if you are actually allowed to use them in the way you do and if you have to give attribution or if there is a copyleft that has to be met. But this depends heavily on the licenses and your usage.
Next is distribution
How will it be distributed? The same way? A runnable JAR that whoever wants to use will download and run?
Maven is often not a very good way to distribute applications. I'd recommend using a open source host like github, gitlab, etc.
Most of them allow very simple release deployment using releases, which may also contain prebuilt jars or installers. But main deployment artifact in open source is the code.
And your last question:
How is the process of turning a certain application version from a github tag into a jar in a repository (E.g. https://mvnrepository.com/)?
This is a very tech-specific question and is not open source related. However, it can also not be answered without further knowledge of your code and you goal. In general, you use the java compiler and utils, see here.
I hope my answers can help you with your project! Cheers