I just created a new sketch, and added two files - that.h and that.cpp. I added code to all 3 files to define and use a class.
I saved the sketch, as ThatTest.pde and closed the IDE.
I navigated to the ThatTest directory, and drug the .pde file onto the icon on the desktop (created by right clicking on the Arduino.exe file and choosing send to desktop as shortcut). The IDE opened, with 3 tabs, for ThatTest.pde, that.h, and that.cpp.
If you are hiding file extensions (the default behavior for Windows), I suggest that send an e-mail to Microsoft and suggest that they are blithering idiots for making hiding extensions the default behavior, and change to show extensions.
If not, I suggest that you try again to create a new sketch and library (header and source), using the open new tab button.
Is it the case that arduino is more C orientated rather than OO C++?
The programming language for the Arduinos is C++, which is a superset of C.