You're right of course that conflicting interests need to be balanced, and the concept of human rights isn't straightforward.
Without wishing to sink too deeply into this (the language of rights isn't normally my preferred way to think about these issues), certain 'basic' human rights such as the right not to be killed or the right not to be tortured can be considered universal and pretty much without exceptions, as long as that person hasn't acted in a way that means they are no longer entirely innocent. The subject of war then comes up, but the targetting of civilians is considered a war crime by many, for this very reason, and plenty of people consider things like the bombing of Hiroshima or Dresden to be war crimes.
But within a scheme where the interests of other animals can be trumped by the interests of humans, I can't see how any kind of universal system of the sort Jeff R espouses can work. At best, you would need a two-level system: human rights, and a set of weaker non-human rights. And this then looks very much like an animal welfare system recast (shoehorned?) into the language of rights.
I basically agree with what you say about human rights, but would add a few qualifiers. First, it is true that most regard the *targeting* of civilians as a war crime but opinion is much more divided about whether the 'collateral' killing of civilians can be justified. And in other circumstances people seem willing to accept the killing of innocent individuals as a side effect of producing a greater good. For example, in the famous 'trolley problem' thought experiment, the vast majority of people are prepared to divert a train onto a side track that will kill an innocent person in order to save the lives of 5 others on the main line. In other words, there seems to be fairly widespread acceptance that in some circumstances it is permissible, and indeed justified, to kill innocent humans.
I also think the concept of 'innocence' needs unpacking. The idea of being innocent as it is used today usually means somebody who is not morally responsible or culpable of wrongdoing, but when one looks at the term's etymology it refers to something different: somebody who is not causing harm (from the Latin in- ‘not’ + nocere ‘to hurt’). You can have individuals who pose threats of harm who are not morally culpable in anyway: people labouring under serious mental delusions, people acting under extreme duress, sleep-walking people, toddlers who have come by their parent's gun etc. Many people think that if these individuals are posing lethal threats or other types of serious harm to others and the only means of averting such threats is to kill them, then killing them can be permissible on the grounds of self-defence, notwithstanding the fact that the attackers are not morally culpable in any way.
So, in my understanding of human rights, there are some instances where it is permissible to kill innocent humans as a side effect and some instances when it would be permissible to kill a non-culpable attacker. There are also circumstances where I think it is permissible to kill innocent animals as a side effect and to kill animals who non-culpably pose threats to us. I accept that farming practices will involve the killing of wild animals (both deliberate and accidental) - and that doesn't contradict my belief in animal rights, any more than my acceptance of the permissibility of killing innocent or non-culpable humans in some instances contradicts my belief in human rights.
Of course, both side-effect killings and self-defence killings are subject to necessity and proportionality considerations: if there are reasonable alternatives to killing they should be used and the killing can only be justified if the harm averted is proportionate to the harm inflicted (and significantly greater with side-effect killings). I take it that many crop-farming practices do not satisfy these criteria and so I would want them to be reformed in ways that do, but I do think that feeding of humans and preventing the spread of diseases like weil's are sufficiently weighty to justify some killing if necessary.
But this is miles away from animal agriculture - which involves the breeding, mutilation, exploitation and killing of animals who pose *no* threat to us and whom we are using not out of any necessity but rather for trivial and selfish reasons related to desire, tradition and convenience.