As for the "dt" notation; it seems problematic but if thought of geometrically, it's a volume form on the real line R; choose the coordinate function "t" and then apply the exterior differential operator to it and we get a 1-form on the real line. This fits well when generalizing to calculus on manifolds. By the way, dt is an exact form by definition but in fact, any 1-form on R is exact because it's 1st cohomology vanishes due to non-compactness.