Is there a connection between depression and personality?

Good question. They are related on several levels. Personality disorders typically involve symptoms of depression but are classified otherwise. Depressed people tend to be pessimists, a personality trait. Although the terms are not correlated: depression is a neurobiological disorder; personality is an individual trait. Both are inherited, but most likely from different genes.
Yes, at times. Officially there is a depressive personality disorder diagnosis being studied for inclusion in the list of accepted personality diagnoses. The fact that there are personality styles we see clearly living in a chronic state of mild depression has always been known. That type of depression doesn't often benefit from antidepressant medication, but can respond to various types of counseling.