Unit testing in spring mvc

If I want to unit test my dao classes in spring would I just call my service methods and test those or would you test the service methods separately to the actual dao methods?

Also should I mock the dao calls or actually use an in memory database like H2? I see that as more of an integration test although some tutorials do this, or would a standard approach be to test with mock database objects for the service tests and use H2 when testing the dao calls?

Finally.. My application has a rest API which is called from the web front end using the Spring rest template and so only the API web app accesses the database.

Would I test the rest methods in each web app using mocked objects and then Start a tomcat instance and integration test between the 2 apps? If I used tomcat and ran integration tests between the apps would connect up a database or mock objects in the API app?

Testing the rest calls from the web app relies really on how the API app's rest method responds so is this even worth testing in isolation?

I find unit testing quite confusing as some of it seems almost to be integration testing.

Does it matter if you run integration tests against H2 in memory but then in reality I would be using MySQL?