Animal brains, especially mammals', are quite similar to humans. So far, attempts to define what makes human cognition "different" in some way have been for the most part a failure - the goal posts continue to be pushed. What seems to remain is a suggestion that human brains are especially socially-focused, and that humans have a uniquely developed ability to understand, plan, and simulate how others may be thinking (for example: Adolphs, R. (2003). Cognitive neuroscience: Cognitive neuroscience of human social behaviour. Nature Reviews Neuroscience, 4(3), 165.).
However, we don't even need to look to apes, primates, or even mammals, to see cognitive abilities of the types that you mention.
For example, at least some birds are capable of quite a lot. As far as easily digestible material, there is a NOVA program on animal minds. I have only seen the bird episode, so I cannot speak to the others.
Although for the made-for-TV format a lot of the details are omitted for the sake of brevity, the program attempts to test the hypothesis that birds are capable of some advanced behaviors that are not attributable to simple memorization or repeated tasks. These include the ability to solve novel puzzles, and to learn hierarchical relationships of the form: X>Y and Y>Z therefore X>Z.
Some others from a variety of species:
Monkeys learning the value of tokens:
Addessi, E., Crescimbene, L., & Visalberghi, E. (2007). Do capuchin monkeys (Cebus apella) use tokens as symbols?. Proceedings of the Royal Society of London B: Biological Sciences, 274(1625), 2579-2585.
Basic arithmetic in chimpanzees:
Rumbaugh, D. M., Savage-Rumbaugh, E. S., and Hegel, M. T. (1987). Summation in the chimpanzee (Pan troglodytes). J. Exp. Psychol. Anim. Behav. Process. 13, 107–115.
A paper and further discussion on generalization of a learned rule in birds:
Kamil, A. C. & Jones, J. J. 1997. The seed-storing corvid
Clark’s nutcrackers learns geometric relationships
among landmarks. Nature, 390, 276–279.
Biegler, R., McGregor, A. & Healy, S. D. 1999. How do
animals ‘do’ geometry? Animal Behaviour, 57, F4–F8
Kamil, Alan and Jones, Juli E., "How do they, indeed? A reply to Biegler et al." (1999). Avian Cognition Papers. 12.
Macaques recognizing themselves in the mirror
Rajala, A. Z., Reininger, K. R., Lancaster, K. M., & Populin, L. C. (2010). Rhesus monkeys (Macaca mulatta) do recognize themselves in the mirror: implications for the evolution of self-recognition. PloS one, 5(9), e12865.
Monkeys reject unequal pay
Brosnan, S. F., & De Waal, F. B. (2003). Monkeys reject unequal pay. Nature, 425(6955), 297.
And I could go on and on - these are just a few examples, not necessarily the best or most current ones. What you will see that all these works share is that it is very difficult to probe animal behavior, because it is often difficult to rule out alternative strategies as we impose our own cognitive biases on animal behavior.
That said, it seems very unlikely that most of the higher-order cognitive skills humans possess are in any way unique to humans. Some specifics like the exact structure of human language are probably fairly specialized, but other social animals also have forms of language, it may just not be as complex or developed as human language.
I'll close with what I find to be a couple cartoons that are highly insightful in this area of research: