The expression
$\frac{\partial f}{\partial t}=\nabla \cdot {\mathbf F}-{\mathbf A}\cdot \nabla \varphi$
(or any similar expression involving vectors and differential operators) is the same in all coordinate systems. But is this true only in orthogonal coordinate systems? Or is the above expression valid in any non-orthogonal coordinate system?
It is valid in any coordinate systems. Actually, any physical laws should be independent of coordinate systems you use. Thus they must be formulated in ways like you proposed (the subject dealing with these stuff is tensor analysis). In introductory courses and textbooks, orthogonal coordinate systems are usually preferred for simplicity.