Is a statically-typed full Lisp variant possible? Does it even make sense for something like this to exist? I believe one of a Lisp language's virtues is the simplicity of its definition. Would static typing compromise this core principle?
Yes, it's very possible, although a standard HM-style type system is usually the wrong choice for most idiomatic Lisp/Scheme code. See Typed Racket for a recent language that is a "Full Lisp" (more like Scheme, actually) with static typing.