Intrinsic definition of arc length There's the Cauchy-Crofton formula for the arc length of a curve. It does not use a parametrization. It also generalizes to formulas for surface areas, etc. en.wikipedia.org/wiki/Crofton_formula The Crofton formulas also have versions that work for parametrized curves (arc length) vs. just the image/trace of the curve.

Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected? @YCor: this is smooth immersions up to 1-parameter families of immersions. Sometimes this is called "regular isotopy" to distinguish it from "isotopy". We're talking about the homotopy-type of the space of immersions. Immersions up to plain isotopy have an enormous amount of components, primarily indexed by the image (as a stratified space).