Let $\mathcal{A}$ be any C*-algebra. Suppose $x\in\mathcal{A}$ is idempotent, with $x\neq 0$ and $x\neq 1$. Does it follow that $\mathcal{A}$ admits nontrivial projections?
Clearly, when $x$ is normal, $x^*x$ is self-adjoint and idempotent, so that settles the matter. I didn't manage to prove the claim in the general case, though.
(for good measure: a projection is a self-adjoint idempotent element of $\mathcal{A}$)
In a unital C*-algebra, every idempotent element is similar to a projection in the algebra. If $e$ is an idempotent element, then $p=ee^*(1+(e-e^*)(e^*-e))^{-1}$ is a projection in the C*-algebra generated by $1$ and $e$, and it is similar to $e$.
For more information see the MathOverflow question Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one. In the question a method that would lead to finding $p$ is outlined. In the answer are further references to this result and generalizations.