Oh yes! Love, loyalty, admiration, frustration, dignity, depression, sadness, happiness, playfulness. Some even have a sense of humour!
I have experienced every one of these from my cats, present and past, and from dogs and birds I've had.
When taking in a new cat, there is always the fear of a new household, but there is a sense of depression/mourning/loss from their old life and home. I experience this when I lost the brother of my first two cats. Freida mourned for almost 3 months! Taking in Tippet, a mutt, she mourned the loss of her previous family and it's - her - children, for about the same amount of time.
Max gets really pissed when we leave for even a short over-nighter. He'll start sulking as soon as the suitcases come out. Then when we come home, he stays attached to me for DAYS. He's getting elderly and needy in his old age.