"Is there c# open source projects team? Where?"
Yes, if you're not planning to start your own project you need to find a community what aligns with your interests. There are plenty of Open Source C# projects, most of them are named something like [projectName].NET.
If you aren't familiar with Linux and MonoDevelop, you may need to be. Open Source projects are usually developed in Linux and made to work on all platforms. It's just the nature of Open Source. Fortunately, the MonoDevelop IDE is very good.
"How can I offer my idea / code
You don't. Chances are, in the larger scope of the project, your idea is insignificant. Keep in mind that, on Open Source projects, the other developers aren't there to work as your personal employees. They're providing a valuable service for free and likely already have a plan of what they want to pursue.
If you have an idea, the absolute first thing you should do is submit a feature request. If your idea is deemed useful by the development team then you can start development. But don't start writing code assuming it will automatically be integrated into the project because you'll likely be wasting your time.
Also keep in mind that your personal style preferences may not match what the project uses. If it's a stable enough project, they may have some coding guidelines posted somewhere.
Either way, whatever you submit will be critiqued and you'll most likely be asked to revise further before it's included in a release.
"If the groups exist ... should I be the project manager in the sense of offering the scope , versions etc. or let others do it?"
If the group exists, you don't 'let' anybody do anything. You don't automatically get benevolent dictator status because you think you deserve it.
If a project is established and development is stable, they will most likely already have a list of feature requests, roadmap for feature inclusion, and a pretty good support team for dealing with bug fixes.
Not just that but, Open Source developers put a lot of time and effort into their work. Not only that but you'll probably be working at some point with people who are much more skilled than you.
Gaining their trust is a slow process, and you will only make progress if you put the same amount of skill and effort into what you offer.
You will likely start with submitting patches, followed by submitting pull requests if your working on something bigger. Don't expect write-access to the main repository because. Contrary to popular belief, Open Source code is closely scrutinized before being merged into the main repository and even then, the submissions will likely fall on a pre-planned release schedule anyway.
Just from your initial approach I'd say, learn some humility before you attempt to join an Open Source project. They're not going to willing hand over the reins just because you think you're a talented developer.
It takes time to earn trust and if you're good you'll eventually earn the respect of the other contributors.