Does the Marshallian demand function always include prices and income?
These are proper Marshallian demand functions, even though Income does not appear in them. This is due to specific form of the utility function (and the candidate solution of all goods being purchased at strictly positive quantities). It emerges that there is no income effect for goods $x_1$ and $x_2$ - optimal uncompensated demand does not depend after all on the level of income, but only on the relative prices.  

The reason why Marshallian demand is defined as it is, is to make clear that it does not include any kind of "income compensation" as Hicksian demand does. But this does not preclude a case like yours, which again, depends on the form of the utility function.  

Thinking economically, what goods can you think of whose demand may realistically not depend on the level of income but only on relative prices? Answering this question would be useful in order to map the mathematical expression of the utility function to real-world economic phenomena.

Also, note that the specific utility function points to "quasi-linear" frameworks, where the good that enters additively, essentially functions as Income itself.