Yes, pets are great that way. A great way to calm yourself is to hang out and pet a cat while s/he is purring and feel the vibration. One way to show your cat that you love her is with a treat. Mine love feline pill pockets by Greenies (rolled up into a ball as a treat, no pill inside). They're great for giving medicine too if you should ever need to one day. ^_^ I'm glad your girl is there for you; hang in there! ^_^
Same thing happens to me- whenever I feel down my cat comes up and rubs herself against me, showing affection. I feel like they can sense your emotions, and just want to make you feel better (: It's really sweet!