Yes. handedness (or Behavioral Lateralization) has been documented in numerous vertebrates (mammals, reptiles and birds) as well as invertebrates.
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, "handedness" (or pawedness, footedness, eyedness, earedness, nostriledness,
toothedness, breastedness, gonadedness, etc.) occurs rather extensively across the animal kingdom.
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.
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. & Vallortigara, G. (1996). Right-pawedness in toads. Nature, 379, 408.
Bisazza, A., Rogers, L. J. & 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. & Vallortigara, G. (2001). Lateralization of ventral fins use during object exploration in the blue gourami (Trichogaster trichopterus). Physiology & Behavior, 72, 575–578.
Bradshaw, J. L. & 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., & Davis, M. (1938). " Left-Handedness" in Parrots. The Auk, 55(3), 478-480.
Hoso, M., Asami, T., & Hori, M. (2007). Right-handed snakes: convergent evolution of asymmetry for functional specialization. Biology Letters, 3(2), 169-173.
McGrew, W. C., & 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. & Rogers, L. J. (1998). Lateralized agonistic responses and hindlimb use in toads. Animal Behaviour, 56, 875–881.
Rogers, L. J. & 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., & 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. & 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. & Vallortigara, G. (2001). Lateralization of response to social stimuli in fishes: a comparison between different methods and species. Physiology & Behavior, 74, 237– 244.
Vallortigara, G., Rogers, L. J., Bisazza, A., Lippolis, G. & 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. & Bisazza, A. (1999). Possible evolutionary origins of cognitive brain lateralization. Brain Research Reviews, 30, 164–175.
Wells, D. L., & 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.