Any thoughts on how that topic? Is 'zombism' a universal problem, or a genetic anomaly (i.e. only affecting related species)?
It depends in part whether the plague is magical (actual returning from the dead), functionally magical (alien origin, etc, where it doesn't play by normal rules), or trying to be more scientific like a virus. Note that saying it's a virus is irrelevant if you treat it in a magical way; I'm referring to actually treating the zombie plague in a somewhat realistic manner.
If the plague is a virus, species limitation is easy: viruses are rarely cross-species, and cross-species jumps tend to burn out quickly. Viruses are just too specialised to be a wide-spread multi-species problem, at least in most cases. This is even more true if the virus was created or changed in a lab. However, there are some viruses which are multi-species, and viruses aren't the only plague vector. A parasite is actually a good choice for a zombie plague since it's basically a small animal that has taken over the body for its own purposes.
In Mira Grant's "Newsflesh" series, the virus' ability to cause zombies was based on how many virus bodies an organism had, and the cut-off seemed to be large dog size. Basically every mammal is infected, including the surviving humans, but the disease is passive unless triggered by exposure to the active form (getting too close to zombies) or by the virus overcoming the drugs in your system. That series also treated zombies more as a hive mind species; the more zombies in an area, the faster and more intelligently they all moved. One zombie can be ignored, a few zombies can be outrun, a pack in a major threat.
the few survivors 'triumph' over the zombies by outliving them by a variety of means, since the zombies lack the wherewithal to maintain a food supply.
In John Ringo's recent series "Black Tide Rising", the human survivors are mostly an ocean bound force, since human population pre-fall means that even secure land bases are so overrun that extraction is impossible without air support. Zombies are discovered to have two unpredicted food supplies: rats and slower zombies. Additionally, zombies tend to hibernate when lacking food, so they can survive longer. The only thing a zombie really needs for survival is a semi-fresh water supply -- and they're not picky about freshness.
One reason that I mention the above series, and I was getting at this with my first post, is that zombies as an actual direct threat has basically burned itself out as a sub-genre. If you're not doing something new and interesting, either with the zombies or with the survivors, then readers no reason to read your story over the thousands of other generic zombie stories. The zombie fiction that's going to survive the next few years will be the stuff that breaks out of the traditional tropes.
Since the zombie is dead, there is no blood flow, so the body gets no food and would not be able to function as described in the typical zombie stories.
Once you assume magical origin then all normal rules go out the window. Reanimated corpses probably don't actually require food, when they "eat" brains or flesh they are actually ingesting a magical life energy or soul.
In the above-mentioned series by Ringo, he gets around the problem by making the plague an actual engineered virus, and a major plot-line in the first part of the first novel involves explaining how it actually works. In short, it's a modified rabies virus with a second viral load spliced in. Rabies acts as a highly contagious airborne vector with flu-like symptoms which gives the blood pathogen (the actual zombie virus) time to establish itself in the host system.
I've heard of some moving from animals to humans, but don't know of any that go from humans to animals - it probably could happen, but not heard of any.
I suspect it probably happens a fair bit, but there are two reasons we wouldn't hear about it. First, most animals with close contact to humans are food supply, so they're not going to last long enough for a disease to take hold; and second, we don't exactly have wide-spread reporting on non-human diseases, so you'd pretty much have to be in an affected field to hear about it.
I did find an article on LiveScience about famous species-hopping diseases. Apparently any modern feline which develops ulcers is due to an early large cat dining on human entrails and contracting Helicobacter pylori, which still persists in lions, cheetahs, and tigers. A couple of respiratory diseases that kill human infants in developing countries wiped out huge populations of chimps in West Africa in 1999 and 2006, probably transmitted due to ecotourism.
It's also thought that primates have caught several diseases from humans, including Polio (chimps in Tanzania), Yaws, which is a non-STD variant of syphilis (gorillas), and anthrax (chimps and gorillas in West Africa). However, it may just be that these diseases exist in jungle conditions and that both humans and apes contracted them from the same source, so it only looks like human transmission because we didn't used to study the apes as closely as we do today.