Do non-human animals also do so, or do they breathe through both nostrils at once? Has any actual study been conducted on this topic?
The post linked by @Remi.b says
The nasal cycle is a natural ultradian cycle (see here and here. Not only is it present in humans, the nasal cycle has been observed in rats, rabbits, domestic pigs, cats and dogs (see references in Eccles 1996).
Proceeding to Eccles 1996 (p. 372):
The nasal cycle is not only limited to the human nose, as it has been found in the rat and rabbit , the domestic pig [32, 33], the cat  and the dog , and appears to be a universal phenomenon at least in all mammals and possibly other animals. ...
- Bojsen-Moller F, Fahrenkrug J. Nasal swell bodies and cyclic changes in the air passages of the rat and rabbit nose. Anat 1971; 110: 25–37.
- Eccles R. The domestic pig as an experimental animal for studies on the nasal cycle. Acta Otolaryngol (Stockh) 1978; 85: 431–436.
- Campbell WM, Kern EB. The nasal cycle in swine. Rhinology 1981; 19: 127–148.
- Bamford OS, Eccles R. The central reciprocal control of nasal vasomotor oscillations. Pflügers Arch 1982; 394: 139–143.
- Webber RL, Jeffcoat MK, Harman JT, Ruttimann UE. Demonstration of the nasal cycle in the beagle dog. J Comput Assist Tomogr 1987; 11: 869–871.
A Pubmed search for '"nasal cycle" animal' gets a total of 18 hits, including refs 32 and 35 above, as well as
- Spontaneous nasal oscillations in dog. A mucosal expression of the respiration-related activities of cervical sympathetic nerve. Asakura K, Hoki K, Kataura A, Kasaba T, Aoki M. Acta Otolaryngol. 1987 Nov-Dec;104(5-6):533-8.
- Proceedings: Studies on the nasal cycle in the immobilized pig. Eccles R, Maynard RL. J Physiol. 1975 May;247(1):1P.
A bit more digging forward & backward through citations finds the article on cats (also by Eccles):
R. Eccles & R. L. Lee (1981) Nasal Vasomotor Oscillations in the Cat Associated with the Respiratory Rhythm, Acta Oto-Laryngologica, 92:1-6, 357-361, DOI: 10.3109/00016488109133272
You might find more if you poke around (Google scholar/Pubmed, look at citations backward & forward, try search terms like "'nasal cycle' bird" or "'nasal cycle' reptile"), but at this point I doubt that anyone's bothered to check this in non-mammalian animals ...