Morality, meaning what is moral and immoral, is defined in the dictionary as being what is truly right and truly wrong. This is based on tangible effects and observations, not mere opinion. However, a huge chunk of society uses the words "moral" and "immoral" when speaking about things which they do or don't personally agree with in their beliefs or personal opinion. One example would be the bikini; while many people love it and there is very clearly no harm to wearing one or seeing someone wear one, those who are more prudish will declare bikinis to be "immoral" because they don't like them. The current top answer by "Jeremy" states, "Morality is decided by the people." That's exactly the fallacy I'm talking about; what is and isn't acceptable by the people changes all the time (remember when non-heterosexuality was considered one of the most immoral things in the world?), while real tangible effects of something to determine whether it is beneficial or harmful, right or wrong, is constant.
Bestiality is defined as sexual activity between human and nonhuman animals (ie. interspecies sex that specifically involves a human). You're very correct in your ending statement that, like any sexual act [with any combination of partners] it has the potential to be wrong, ie. immoral, if there is physical or psychological harm. Also, if it is done without the full consent of everyone involved of all species.
To address the points you referred to, the "unnatural" argument is false because interspecies sex is an observable fact of nature that has happened between countless species throughout time. We get ligers from a male lion and a female tiger (and tigons are the offspring of the opposite combination of genders); we get mules from a horse and a donkey having sex; we get hybrid plants from two different species combining genes to make a third type of plant. And it's not limited to physically similar or genetically similar creatures; I've read of mateships (emotional and/or sexual) between a goat and a sheep, a goat and a dog, a dog and a cat, a cat and a raccoon, and many different pairs of birds of different species. It occurs in nature with many different animal species, humans included, and among the human species has happened since at least the earliest recorded history. There are statues thousands of years old of humans having sex with bears, dogs, etc., ancient written history about it happening, and if I remember right there are even very ancient cave paintings depicting a human having sex with some species of hooved animal.
The "animals can't consent" argument is false because simple observation will show that animals decide and communicate consent to other animals (humans included) about all sorts of things all the time. They can communicate consent to a tummy rub, rejection of someone playing with their feet, consent to a car ride, rejection to a bath or visit to the vet, and so on. It's stupid to think animals can't also decide acceptance and consent, or rejection, to a sexual suitor of the same or a different species. Watch any nature show and you'll see animals having sex. When they try to win mates and want to have sex they show it with vocalization and body language (nonverbal communication) and the one receiving the offer can accept or reject the offer and communicate that message non-verbally. Stance, posture, peeing, spread legs, tails off to the side, erections, and so on all say "Yes"; moving away, sitting or laying down, growling, flattened ears, teeth, claws, and hooves all say "NO." It would be no different with a human; if the animal offers or requests sex, it's done through nonverbal communication for the human to interpret. If the human is the one offering/requesting, the animal receives that message, decides consent, and communicates back whether he/she accepts or rejects that proposition. It's then up to the human to receive, understand, and obey that message, especially if a rejection. For the human to not see and understand that communication, he/she would have to be ignorant and stupid to the point of blindness. For the human to ignore a lack of consent (or never look for it to begin with) and proceed with sex anyway whether through use of coercion, threat, punishment, restraint, or sedation, that would very clearly be rape. Rape is rape regardless of the species or gender of the victim and should always be very severely punished.
It's a poor comparison to put sexual consent up against slaughter consent. It should be very obvious that no animal in the world is going to consent to being killed and eaten. Humans ignore that non-consent for being killed all the time. To compare that to sexual consent could be interpreted as saying, "we ignore their desire to live and kill them anyway, so what's the problem with ignoring their lack of consent to have sex?" The matter of sexual consent/activity needs to be proven and stand under its own merits.
Back to the morality issue. If two beings, in this case a human animal and a nonhuman animal, decide to have sex together, they are experiencing and giving each other great pleasures and sexual release. It involves no one else, they are both willing participants, they're both having a good time, and no harm is done to either one. With both benefiting from such enjoyment that could be considered to be right and therefore moral based on the definition of moral (even if people would object to the idea and call it immoral in their opinion anyway). If those two beings have sex without a total mutual consent, or the human forces sex on the animal in some way, or the animal's right and/or ability to refuse is taken away, or the human is forced into doing it by some third party, or either one receives any sort of mental/emotional trauma, discomfort, pain, injury, suffering, or death due to a major anatomical size mismatch or physiological incompatibility (ie. man having sex with chicken, which is always fatal due to the physical nature of the cloaca), then due to that real harm the sexual activity would be truly wrong and therefore rightly defined as immoral.
BTW, there's really no difference between mounting your dog (being the giver) and being mounted by your dog (being the receiver) because it is sexual activity with your dog in either case, and being the receiver you could just as easily refuse to allow it to happen as in being the giver.
The reason people declare bestiality, or the relationships held by those whose sexual orientation is zoophile/zoosexual, to be immoral is because that's what they've always been taught to feel by society, their religion, peer opinion, etc. With such distaste for the topic they never think to look into it themselves to understand the activity, the people, and the motivations in order to form a more valid judgment about it. How many people declared the Harry Potter series of books to be immoral without even cracking a single volume to read one page? Millions. Some even chose to burn them. Yet if they'd read it to see what it's all about they would have seen that the books are fun and well written, tell a story without trying to teach anyone to believe that the magical stuff is true, and that their objection was over nothing. People like that would rather feel righteous about themselves and cast a loud negative judgment over things they actually know little to nothing about.