Many yes. They can suffer from depression, OCD behaviors, aggression, ( conduct disorder lol ) They can mourn, fail to thrive and ultimately die of emotional stress such as geese can fall depressed and die upon losing a long time mate. My goose was on anti-depressants after his mate of 10 years died and the vet was worried about him.
I do know there are not any sociopathic or narcissistic animals except man. haha
Of course animals can suffer mental and emotional health problems based on nature/nurture just like us.