A paracompact manifold admits a Riemannian metric.
Does this hold for a manifold with boundary?
If not, then under what conditions does it admit a Riemannian metric? Say, is it true for a compact manifold (with boundary)?
Not a duplicate: In this question, it is said that the metric can be extended from the boundary; however, the question assumes the manifold to be already Riemannian, while my question is whether any (say, para compact) manifold with boundary can be made Riemannian in the first place.
Yes, there's always a Riemannian metric on a manifold with boundary. With the usual partition of unity construction, you get a Riemannian metric on the double of the manifold, so just restrict it. [Or take a two-sided tubular neighborhood of the boundary to extend the manifold beyond the boundary, and then put a Riemannian metric on that.]