My question is, do these formulas always hold?
The answer is really "yes" and "no". Other answers have explained the "yes" answer, but they all depend upon a capacitor or inductor being "ideal". Real capacitors and inductors have "stray reactances" and resistances. But even if we ignore these, real capacitors have dielectrics which are not a vacuum (although air comes close). Real inductors have cores which are not a vacuum (although air again comes close).
The significance of these facts is this:
The reactance of a real capacitor will deviate from that of an ideal capacitor, and that deviation will depend upon both frequency and upon amplitude. The permittivity of every real non-vacuum dielectric is non-linear, (although air comes close).
Similarly, the reactance of a real inductor will deviate from that of an ideal inductor, and that deviation will depend upon both frequency and amplitude. The permeability of every real, non-vacuum core is non-linear (although air comes close).
Everything that is said above about the reactances deviating from ideal applies also to the differential equations that govern ideal capacitors and inductors. Real components will behave differently from the differential equations for ideal capacitors and inductors
\$I = -CV'\$
\$V = -LI'\$
even if stray inductance, capacitance and resistance are accounted for.
Design objectives in practical power circuits generally include minimizing bulk, weight, and cost. Unfortunately, these objectives conflict with linearity of components. Inductors with magnetic cores are highly non-linear, but are used in power circuits because they are smaller, lighter and cheaper than their more linear equivalent counterparts. Similarly with capacitors. In practical circuits where these components are used near their voltage or current limits, their reactance may differ quite significantly from the values obtained with small signals. Power supply engineers generally need to take the non-linearity of their reactive components into account.