Hi,
Most of the questions here can be answered by following the Gemfire online tutorials and introductory documentation - vFabric Documentation Center
That said, let's follow up your questions:
1 - If you're already using some ORM framework like Hibernate and JPA, Gemfire can easily be used as L2 Cache, which will do exactly that for databases. But if you're not using Hibernate or JPA, or not even a database, you can implement your own Data Loader (vFabric Documentation Center) that will actually load data when a cache miss occur.
2 - GemFire is product itself, with it's own API's. Spring Data is a framework that provides an API for access to a range of systems, including GemFire. It's important to notice that Spring & Gemfire are both under Pivotal's umbrella, and that's what is most important here. 
3 - Eclipse or STS is fine.
4 - Yes, Gemfire can be used within any application server, although it provides support for HTTP Session replication for the most relevant ones. You can check the documentation for more details.
5 - Don't know, you should probably post some error information and if it's actually Spring data related, there is a specific forum under SpringSource website.
6 - Please check GemFire tutorials and look for client/server examples.
Hope that helps,
~/William