Market demand independent of distribution of income
I take homogeneity of preferences here to mean that preferences are homothetic; that is, preferences can be represented by a utility function that is homogeneous of degree one. 

For an arbitrary good $x$ and a vector of prices $\mathbf{p}$, let individual demand be $x_i(\mathbf{p}, M_i)$ for each consumer $i$ with income $M_i$. Thus market demand is $D = \sum_{i} x_i(\mathbf{p},M_i)$. Since preferences are identical, 

$$\sum_{i} x_i(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},M_i).$$ 

Then, by homogeneity, 

$$\sum_{i} x(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},1) M_i = x(\mathbf{p},1) \sum_{i} M_i = x(\mathbf{p},1)M,$$ 

where $\sum_i M_i = M$ is aggregate consumer income. Thus $D$ is independent of the distribution of income and is solely a function of aggregate income across consumers, $M$.