>If I have Mathematica, does that allow me to execute code against the giant corpus of Wolfram Alpha knowledge?
Yes. you can use results of natural inputs as a computable data.
For example, type ==dog picture, this will give you exactly the same result as Wolfram|Alpha. then you can click on "+" sign in the top right corner of the pod and select "subpod content". which will give something like
WolframAlpha["dog picture", {{"Image:SpeciesData", 1}, "Content"}]
and it is an image you can do processing on.