So I did a lot of research on this topic and want to share my findings with you:
One (from a technical point very attractive) option is to use commercial bridges between Java and .Net. For sure, the most popular products are JNBridge and Javonet. Both products seem to be quite easy-to-use, have good support and seem to be very sophisticated. Especially JNBridge already supports bridging between Java and Mono too, which allows the portation to also non-Windows OS, which is one of our main requirements as stated above. Javonet also wants to integrate Mono and is going to release this feature soon. However, both solutions are commercial and one needs to weigh their features against the respective costs. Nevertheless, from a pure technical point of view, they look great and also state to enable very fast communication between Java and .Net (faster than with web services).
Another option is to connect Java and .NET via COM. Since COM is generelly defined platform-independently, this could work on multiple OS. There are lots of open source projects that could be used for such an implementation, such as EZJCOM, J-Interop, JACOB or JCOM. The main restriction (expecially for our project) is that Mono only supports COM-interoperability under Windows (yet). So, this is not really an option for us. But if you want to create Java-.NET interoperability on Windows only, this is a good way.
The straighforward way of integrating Java and C# is to use Java Native Interface (JNI). You can also find manifold implementations that make JNI more easy to use, the most popular one is probably jni4net which seems to be a very active and frequently used project. But there are also others with specific pros and cons, such as Caffeine, Espresso or csjni. Finally, JNI is not 100% platform independet. It is applicable on different platforms, but you have to generate platform-specific code which makes it clearly less usable for our purposes. If you limit your application to Windows, jni4net seems to be a very good choice.
The third option could be to run both the Java and the .Net part within a Common Language Runtime. Ikvm.net is one possible and very popular solution therefore (as mentioned above by Samuel Audet). The drawback of this option is the loss of features and efficiency of the JDK.
The last and surely most generic alternative is to set up webservices between the Java and the .Net world. For this solution, one needs to find appropriate ways for serializing/deserializing objects from/to Java and .Net. There are manifold possible solutions for that available. RenniePet mentioned a sophisticated solution based on Protocol Buffers. Others exist as well such as http://java-cs-bridge.sourceforge.net/. This option might have a potential drawback when considering communication runtime, but may be the way to go for us.
Hope this may help anyone in the future that is confronted with the same problem.