Flame zones. Wax does not burn as a solid. The heat melts it, the wick soaks it up, the hot tip turns it to vapour, and the vapour burns. In the dark middle there is vapour but almost no air, so it does not burn (about 600 °C). At the bottom the air mixes in and the flame glows blue (about 1400 °C). In the yellow part, tiny soot specks get white-hot and shine; that is the light you see.
Wick (capillary rise). Liquid climbs thin gaps because it sticks to the fibres. Jurin height: h = 2γcosθ / (ρgr) (γ surface tension, ρ density, r gap radius). On the way up the speed is limited by thick liquid (Washburn): h = √(γr cosθ t / 2μ) at the start. The wax flow used here: ṁ = ρφA (r²/8μ)(2γ/r − ρgL)/L.
Burn rate. The flame makes about 0.9 · ṁ · 46 kJ/g of heat. Warming, melting and boiling one gram of wax needs only about 1.1 kJ, so about 2.5% of the flame heat is enough to keep it going. Flame height is taken as proportional to the vapour flow (22 mm at 0.1 g/min). Burn demand grows with the wick length as a toy power law; a wick that is too long wants more wax than the wick can bring, so it chars and smokes.
Blowing. A breath is a jet. Beyond about 6 mouth widths its speed falls like u = U·6.2 d / x, so doubling the distance halves the speed. The flame leans by atan(u / 0.35 m/s). If the air is faster than a critical speed (about 2.5 m/s) for long enough (about 0.12 s per unit of excess), it carries away the hot vapour and the flame goes out. This is a simple threshold, not a full combustion model.
Smoke trick. Just after blowing out, the white smoke is wax vapour and tiny droplets. A match held in it can light that vapour, and the flame runs down to the wick. It only works while the smoke is thick (close to the wick, in the first few seconds). Real smoke is messier than the single decay used here.
Jar. Burning paraffin uses oxygen: C25H52 + 38 O2 → 25 CO2 + 26 H2O, that is 0.108 mol O2 per gram. The flame dies when oxygen drops from 20.9% to about 16%. Burn time t = (n₀·0.209 − 0.16 N₀) / (ṁₙ (1 + 0.34·0.16)), with N₀ = pV/RT. The model keeps the burn rate constant and ignores heat and glass effects.
Honest limits. All numbers are rounded textbook values. Real candles differ by wax, wick and room draughts. The dinosaur cake is just decoration.
Running...