Does any other animal (apart from humans) have conscience, feel guilt, and have the ability to discern right and wrong?
Marc Bekoff and Jessica Pierce start the first chapter of their book, Wild Justice: The Moral Lives of Animals with these words:
Let's get right to the point. In Wild Justice, we argue that animals feel empathy for each other, treat one another fairly, cooperate towards common goals, and help each other out of trouble. We argue, in short, that animals have morality.
They admit, "This jump is controversial not for scientific reasons so much as philosophical ones...." (p. 3)
One such concern may be the question the OP asks: "Do other animals have conscience?" In other words, the moral descriptions of animal behavior may be accurate descriptions of correlates of conscience, but just because animals exhibit such behavior suggesting they have conscience does not mean they actually do have conscience.
Since we do not actually feel what they are feeling we could maintain skepticism about their having conscience just as some people remain skeptical about the very consciousness of other human beings.
They describe two positions on conscience. One position in favor of conscience in non-human animals comes from Darwin (p. 145):
In The Descent of Man, Darwin wrote, "Besides love and sympathy, animals exhibit other qualities connected with the social instincts, which in us would be called moral; and I agree with Agassiz that dogs possess something very much like a conscience."
Others argue for a different position (p. 146):
On the other hand, conscience may be something more particular than either impulse control or the internalization of a set of norms about good and bad. Anthropologist Christopher Boehm's work on social sanctioning and origins of conscience in humans suggests a somewhat different answer to the question of conscience in animals. According to Boehm moral conscience is a uniquely human capacity and conscience is an essential component of morality. His hypothesis is that conscience evolved in Homo sapiens in response to the shift during the Middle to Late Pleistocene from subsistence to hunting of large game.
As human beings we are different from other animals since we are a separate species. There must be features which differentiate us from these other animals. Having a conscience may, or may not, be one of those differentiating features.
Bekoff, M., & Pierce, J. (2009). Wild justice: The moral lives of animals. University of Chicago Press.
Yes, they do. There are plenty of examples where cats or dogs suffered after a human's death even if another human provided them with the same food and environment.
Cats wouldn't feel bad about knocking over an object, but they will know they did wrong if you caught them getting food from the table or actually doing something they're not suppose to. As far as cats are concerned they have not done anything wrong at all, but they knew they did it against your will/orders.
Many animals are smarter than people make them out to be. Many can communicate in ways humans can't even imagine. As for cats, they can detect the electromagnetic anomalies in damaged tissue and they will try to repair it. The frequency they generate (~21-27Hz, see Eklund, Peters & Duthie, 2010) when they purr is scientifically proven to be a frequency that accelerates living cell regeneration (James E. Trosko, Michigan State University research, 2000; A. Borg, Direct Effects of ELF; E. von Muggenthaler - Research, the accelerometer tests) . Dogs are good at detecting emotions and if you are friendly with it the dog will try to do something to cheer you up or improve your emotional state. That is if they themselves are not too affected at that point in which case they will be sad, too.
Also, animals seem to have superior collective conscience, because it has been proven that a certain type of animal in an area can adapt in advance to what the same type of animal in another area encountered or experienced without any connection between the two areas (initial proof by zoologist and ethologist Lyall Watson and his colleague Lawrence Blair).