Do animals exhibit handedness (paw-ness?) preference?
Short Answer 
Yes. handedness (or Behavioral Lateralization) has been documented in numerous vertebrates (mammals, reptiles and birds) as well as invertebrates.

This includes domestic cats (see Wells &amp; Millsopp 2009).

 Long Answer 
There have been numerous studies that have documented behavioral lateralization in many groups of animals including lower vertebrates (fish and amphibians), reptiles (even snakes!), birds and mammals. More recent work (e.g., Frasnelli 2013) has also shown that lateralization can also occur in invertebrates. In other words, &quot;handedness&quot; (or pawedness, footedness, eyedness, earedness, nostriledness,
toothedness, breastedness, gonadedness, etc.) occurs rather extensively across the animal kingdom.

These studies suggest that the evolution of brain lateralization, often linked to lateralized behaviors, may have occurred early in evolutionary history and may not have been the result of multiple independent evolutionary events as once thought.

Although this view of brain lateralization as a highly conserved trait throughout evolutionary history has gained popularity, it's still contested (reviewed by Bisazza et al. 1998; Vallortigara et al. 1999).


Note: Laterality of function may manifest in terms of preference (frequency) or performance (proficiency), with the former being far more often investigated.
And no, right-handedness is not always dominant.
But Why?

One hypothesis is that brain lateralization was the evolutionary result of the need to break up complex tasks and perform them with highly specialized neuronal units to avoid functional overlap (i.e., to account for &quot;functional incompatibility&quot;).

In humans, many hypotheses have been developed including: division of labor, genetics, epigenetic factors, prenatal hormone exposure, prenatal vestibular asymmetry, and even ultrasound exposure in the womb.

Snake studies (see below) have suggested lateralization behavior can be dictated by environmental conditions (specifically, temperature).

Other work (Hoso et al. 2007) suggest that lateralization could be the result of convergent evolution. In this case, snakes developed feeding aparati that allow them to better consume more-common dextral species of snails.

Note: dextral (meaning &quot;clockwise&quot;) is a type of chirality -- another form of &quot;handedness&quot;



Reviews:

Lateralization in non-human primates: McGrew &amp; Marchant 1997.

Lateralized behaviors in mammals and birds:
Bradshaw &amp; Rogers 1993; Rogers &amp; Andrew 2002.

Lateralized behaviors in lower vertebrates: Bisazza et al. 1998; Vallortigara et al. 1999.


Some Examples:
Invertebrates

Some spiders appear to favor certain appendages for prey handling and protection (Ades &amp; Novaes Ramires 2002).

Octopi (or octopodes) preferably use one eye over the other (Byrne et al. 2002; with seemingly no preference for right/left at the population level: Byrne et al. 2004) and also apparently have a preferred arm (Byrne et al. 2006).


Fish

Preferential ventral fin use in the gourami (Trichogaster trichopterus). [Bisazza et al. 2001].

Preferential eye use in a variety of fish species [Sovrano et al. 1999, 2001].


Amphibians

Lateralization of neural control for vocalization in frogs (Rana pipiens). [Bauer 1993].

Preferential use of hindlimbs (Robins et al. 1998), forelimbs (Bisazza et al. 1996) and eyes (Vallortigara et al. 1998) in adult anurans.


Snakes

Preferential use of right hemipenis over left under warm conditions. [Shine et al. 2000].

Coiling asymmetries were found at both the individual and population levels. [Roth 2003].


Birds

Tendency for parrots to use left-feet when feeding. [Friedmann &amp; Davis 1938].

Mammals

Pawdness in mice. [Collins 1975].

left forelimb bias in a species of bat when using hands for climbing/grasping. [Zucca et al. 2010]

Behavior experiments show domesticated cats show strong preference to consistently use either left or right paw and that the lateralized behavior was strongly sex related (in their population: ♂ = left / ♀ = right). [Wells &amp; Millsopp 2009].


Non-human Primates

Posture, reaching preference, tool use, gathering food, carrying, and many other tasks. See McGrew &amp; Marchant (1997) for review.


Citations

Ades, C., and Novaes Ramires, E. (2002). Asymmetry of leg use during prey handling in the spider Scytodes globula (Scytodidae). Journal of Insect Behavior 15: 563–570.

Bauer, R. H. (1993). Lateralization of neural control for vocalization by the frog (Rana pipiens). Psychobiology, 21, 243–248.

Bisazza, A., Cantalupo, C., Robins, A., Rogers, L. J. &amp; Vallortigara, G. (1996). Right-pawedness in toads. Nature, 379, 408.

Bisazza, A., Rogers, L. J. &amp; Vallortigara, G. (1998). The origins of cerebral asymmetry: a review of evidence of behavioural and brain lateralization in fishes, reptiles and amphibians. Neuroscience and Biobehavioral Reviews, 22, 411–426.

Bisazza, A., Lippolis, G. &amp; Vallortigara, G. (2001). Lateralization of ventral fins use during object exploration in the blue gourami (Trichogaster trichopterus). Physiology &amp; Behavior, 72, 575–578.

Bradshaw, J. L. &amp; Rogers, L. J. (1993). The Evolution of Lateral Asymmetries, Language, Tool Use and Intellect. San Diego: Academic Press.

Byrne, R.A., Kuba, M. and Griebel, U. (2002). Lateral asymmetry of eye use in Octopus vulgaris. Animal Behaviour, 64(3):461-468.

Byrne, R.A., Kuba, M.J. and Meisel, D.V. (2004). Lateralized eye use in Octopus vulgaris shows antisymmetrical distribution. Animal Behaviour, 68(5):1107-1114.

Byrne, R.A., Kuba, M.J., Meisel, D.V., Griebel, U. and Mather, J.A. (2006). Does Octopus vulgaris have preferred arms?. Journal of Comparative Psychology 120(3):198.

Collins RL (1975) When left-handed mice live in righthanded worlds. Science 187:181–184.

Friedmann, H., &amp; Davis, M. (1938). &quot; Left-Handedness&quot; in Parrots. The Auk, 55(3), 478-480.

Hoso, M., Asami, T., &amp; Hori, M. (2007). Right-handed snakes: convergent evolution of asymmetry for functional specialization. Biology Letters, 3(2), 169-173.

McGrew, W. C., &amp; Marchant, L. F. (1997). On the other hand: current issues in and meta‐analysis of the behavioral laterality of hand function in nonhuman primates. American Journal of Physical Anthropology, 104(S25), 201-232.

Robins, A., Lippolis, G., Bisazza, A., Vallortigara, G. &amp; Rogers, L. J. (1998). Lateralized agonistic responses and hindlimb use in toads. Animal Behaviour, 56, 875–881. 

Rogers, L. J. &amp; Andrew, R. J. (Eds) (2002). Comparative Vertebrate Lateralization. Cambridge: Cambridge University Press.

Roth, E. D. (2003). ‘Handedness’ in snakes? Lateralization of coiling behaviour in a cottonmouth, Agkistrodon piscivorus leucostoma, population. Animal behaviour, 66(2), 337-341.

Shine, R., Olsson, M. M., LeMaster, M. P., Moore, I. T., &amp; Mason, R. T. (2000). Are snakes right-handed? Asymmetry in hemipenis size and usage in gartersnakes (Thamnophis sirtalis). Behavioral Ecology, 11(4), 411-415.

Sovrano, V. A., Rainoldi, C., Bisazza, A. &amp; Vallortigara, G. (1999). Roots of brain specializations: preferential left-eye use during mirror-image inspection in six species of teleost fish. Behavioural Brain Research, 106, 175–180.

Sovrano, V. A., Bisazza, A. &amp; Vallortigara, G. (2001). Lateralization of response to social stimuli in fishes: a comparison between different methods and species. Physiology &amp; Behavior, 74, 237– 244.

Vallortigara, G., Rogers, L. J., Bisazza, A., Lippolis, G. &amp; Robins, A. (1998). Complementary right and left hemifield use for predatory and agonistic behaviour in toads. NeuroReport, 9, 3341–3344.

Vallortigara, G., Rogers, L. J. &amp; Bisazza, A. (1999). Possible evolutionary origins of cognitive brain lateralization. Brain Research Reviews, 30, 164–175.

Wells, D. L., &amp; Millsopp, S. (2009). Lateralized behaviour in the domestic cat, Felis silvestris catus. Animal Behaviour, 78(2), 537-541. 

Zucca, P., Palladini, A., Baciadonna, L. and Scaravelli, D. (2010). Handedness in the echolocating Schreiber's long-fingered bat (Miniopterus schreibersii). Behavioural processes, 84(3): 693-695.