In zero (or micro) gravity there are lots of problems for a birthday cake. Cakes "rise", and flames "burn up", without gravity how could any of this happen?
Will my cake fail to rise? Will my candles burn?
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The cake will "rise" for the same reasons it does in gravity (yeast or other agents releasing gases as a metabolic byproduct, when heated, or as a chemical process and reacting to other ingredients, given sufficient time and providing for other required conditions to activate the rising agent). The only problem is, it will "rise" equally in all directions in a zero gravity environment, so you might want to start shaping your dough well before it starts increasing in volume due to internal pressure upon release of gases, maybe even contain this process completely within a cake shaped container that you bake the dough in (there's no real names for such "kitchen tools" yet for those that will be required in zero gravity, so I'm just using some generic terms - I'm sure Gordon Ramsay would have something to yell about regarding my choice).
And the candles will burn without the flame "rising" in a single direction, but again - equally in all directions, so the "flame" will be more or less spherical, with only the candle wick slightly misshaping it towards a cylinder shape (assuming it will distribute wax due to suction - a byproduct of burning and expanding in volume - equally along its length). So the flame would look like a product of a sphere and a cylinder (slightly flattened "top"), but nearly completely flattened where the candle wax is still solid ("bottom") due to candlestick's surface tension, melting of wax, and suction of air from all directions equally - but the candlestick is slightly in the way and this air pressure slightly increases there as the path narrows). This experiment has been performed on Mir and the video is available on YouTube, and the shape the flame forms confirms all the physical processes described, albeit the experiment seems to have been conducted in a smaller to sea-level Earth's atmospheric pressure, or in atmosphere with slightly smaller percentage of oxygen (the colour of the flame suggests the former to be more probable cause of it).
Candle flame in Zero-G on the Mir
Yes, you could have a cake in zero-g, and you could eat it, but then you wouldn't have it any more. :)