Q:

Is every idempotent density operator the sum of orthogonal projections? 

Can every idempotent density operator be written in the form $$\sum_ic_i|i\rangle\langle i|,$$ where $\{|i\rangle\}$ is an orthonormal set and $\{c_i\}$ is a set of coefficients (Edit: Without assuming that the vector space on which the density operator operates is finite dimensional)? 
It seems like people are assuming that when proving that a density operator is idempotent if and only if it represents a pure state (for example here: Necessary and sufficient conditions for a pure state). 

A:

By definition, density operators are self-adjoint.  
In the case of a finite dimensional vector space, every self-adjoint operator has an orthonormal set of eigenvectors (https://en.wikipedia.org/wiki/Spectral_theorem). 
Alternatively (instead of assuming a finite dimensional vector space), we can prove the statement for compact operators (see exercise 6 in chapter 19 of [1]). 
[1] Brian Hall, Quantum Theory for Mathematicians

