1 :: What does CLIPPER stand for?

► A clipper is a fast or prestigious ship.
► CliPPer = C Plus Plus
► cLIpper = LIbrary
► clippER = ER, thats enough acronym for now.
► Or if you prefer, "Cross-crystal Likelihood Phase Probability Estimation and Refinement, which is what I hope it will be used for.

2 :: Do I need CCP4?

No. CCP4 is an optional dependency which is only needed if you want to access CCP4 file formats.

3 :: Why C++ used in Clipper?

This is where I'm supposed to tell you that C++ is a great language. It isn't (at least not for my purposes). A good programming language allows you to think in the language of the problem, not the language of the computer. In C++ you keep having to return to implementational details, such as whether to use a pointer or a reference or a value, a derived class or a template. This is probably unavoidable if good performance is going to be achieved.
But the main reason is a lack of alternatives. Java is (generally) slow, and clumsy in places. Maybe Objective-C, Delphi, or even C# have solved some of these problems, but they are less portable and less widespread than C++.

4 :: Why use references rather than pointers in the public API, particularly for arguments which are modified?

The C++ provides the most precise statement of the priciples:
No pointers in the public API, except when returning a pointer which might be null. The above reference also describes the problem of wrong attitudes brought over from C.

5 :: Can I use Fortran/C/Python/Visual Basic?

You can provide SWIG wrappers for the parts of your program you need to access.