Is there any evidence that depression or anxiety affect HDL cholesterol levels?

Mixed findings. Many studies have yielded conflicting results, for a variety of reasons. But a larger study from the netherlands in 2010 suggested that melancholic depression was associated with lower hdl. Also, atypical depression was associated with higher total and LDL cholesterol -- hinting that in at least some depressions, lifestyle may be a factor.
Perhaps. Depression is linked to obesity, and the metabolic syndrome. These patients often have high triglycerides and low HDL cholesterol, the former most often being the cause of the latter. This may be more of an association than direct cause and effect.