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The science
Why not just Newton's law? Newton's law of cooling, T(t) = Troom + (T0 - Troom) e-kt, is exact only when the heat loss is proportional to the temperature difference. An open cup of coffee loses heat four ways at once, and two of them are not linear: evaporation grows roughly exponentially with temperature (the saturation vapour pressure of water doubles about every 12 degC), and radiation goes as T4. This model keeps all four, plus the heat capacity of the cup itself.
The equations. Coffee: m c dTc/dt = -G1(Tc - Tw) - Qevap - hcAs(Tc - Tr) - εσAs(Tc4 - Tr4). Cup wall: Cw dTw/dt = G1(Tc - Tw) - G2(Tw - Tr), where G1 = Aw/(1/hin + t/2k) couples the coffee to the wall (hin = 250 W/m²K, wall thickness t and conductivity k of the material) and G2 = Ao/(t/2k + 1/(hc + hr)) couples the wall to the room. Evaporation: Qevap = Lv(T) hm As [ρv,sat(Tc) - φ ρv,sat(Tr)] with the mass-transfer coefficient from the Lewis analogy, hm = hc/(ρacp,aLe2/3) (Le = 0.85), the vapour pressure from the Antoine equation, and hc = 5.7 + 3.8 v W/m²K (flat-plate correlation, v = air speed in m/s). The evaporated water also removes mass, so the coffee level slowly drops. The equations are integrated with a 1 s fourth-order Runge-Kutta step.
The cups. Ceramic, glass and paper differ in wall mass, thickness and conductivity: a heavy ceramic mug first has to be warmed (it steals about 20 % of the heat in the first minutes unless you pre-warm it), while a paper cup has almost no heat capacity but a thin wall. The stainless vacuum flask is not modelled from first principles: its wall-to-coffee conductance (0.05 W/K) and lid conductance (0.015 W/K) are typical assumed values, with a low-emissivity outer skin (ε = 0.1). A paper-cup lid is treated as vapour-tight with a conductance of 3.3 W/m²K; real lids leak steam through the sip hole.
Milk. Pouring 30 mL of 5 degC milk is an instantaneous enthalpy balance (milk specific heat 3.93 kJ/kg K). In a Newton-type cup this makes the answer to the old puzzle exact: milk first leaves the coffee warmer, because the hotter coffee loses heat faster while it waits (check 4).
Simplifications, honestly. The coffee is perfectly mixed (no thermal stratification); the surface is at the bulk temperature (in a real cup the top is a few degrees cooler, which makes evaporation here slightly too strong); the cup stands on a cork coaster with no heat loss through the base; the cup and the room walls are black bodies as far as radiation goes (surface emissivity 0.96 for water); densities and specific heats are constants. The 55 to 65 degC band is a commonly quoted preferred drinking range, not a medical limit. Absolute minutes will differ from your kitchen by tens of percent; the ranking of the cups and the shape of the curves are the point. The four checks below the controls verify the numerics against closed forms.