Yay, a debate I might be able to join in with!
1) Should humans be responsible for regulating animal populations?
As others have said before me, it does entirely depend on the situation, and how the 'control' is being implemented. There are three major situations which I can think of: introduction of a new species, accidental introduction and farming/captivity.
Obviously as ultraviolet has previously stated, release of an animal which functions well in one ecosystem might completely destroy a population of animals. I seriously doubt all of the research in the world could accurately predict an introduced species' behaviour and interactions with its new environment and then you end up with masses of unwanted animals, such as rabbits and cane toads.
Conversely, you have species – such as the rat – which have climbed aboard ships when developed countries at the time were starting to explore the world. Unintentional introduction of species can be more harmful as in these cases, we're not even monitoring what's going on with the ecosystem. The Tuatara is now only found in remote areas because of the introduction of rats. I would agree that eradication of rat populations from these areas where they threaten the conservation of the Tuatara is acceptable for three main reasons: Without human intervention, the Tuatara would probably still be alright in its natural environments. Secondly, rats wouldn't have been introduced without accidental human interference. Lastly, rats are extremely common, so eradicating them in one area is not going to cause them to go extinct.
I think one of the largest areas where population control is highly important, which hasn't been mentioned yet, is in farming. With oversized farm populations, we end up with a 'battery' farming situation of low welfare imposed on the animals. However, at the same time, a farmer has to keep his profit margins within reason and as such, the productive value of an animal throughout its lifetime will impose on its lifestyle – whether it be breeding for replenisher stock, or for slaughter.
Any form of milk production will involve pregnancy. Control of the breeding seasons in production animals is highly important in the maintenance of food production, as a cow will only lactate due to calving. However, this then brings up ethical issues as only female animals produce milk. Though the bull could be slaughtered for beef, many are shot on being born because the dairy breeds do not have a good killing out percentage. However, if farmers had to keep and rear all of their bulls, they would lose a large amount of their profit which could mean lower welfare standards for all of the animals in the system.
Artificial insemination is very common in large production systems for dairy cattle due to low fertility rates. The fertility rates are low due to the drive for productivity – cows with a higher milk yield are less fertile due to a genetic link. As such, the population needs to be controlled this way in order to prevent there being too few cows born (and thus, no milk shortage).
2) Should individual animals be sacrificed for the 'greater good' of an ecosystem, or should all individuals be given equal moral consideration?
It really depends on the circumstances. In most circumstances, I don't agree with killing healthy animals however, if there is a large disruption to the ecosystem caused by animals then it's sometimes necessary. With the Tuatara, it was only due to humans that caused the introduction of rats to the ecosystem, so it seems more justified. We cause the problem, we do our best to fix it and the best way to do that is to get rid of what caused the initial imbalance. However, if the animal was already native to the population, or has migrated to a new location, this is really just nature taking its course. There are probably a few circumstances where removal of a species can be justified, but they're likely rare circumstances.
3) Should certain species of animals be given more rights than others? Should endangered species be given more consideration than non-endangered species?
You would swat a fly but you wouldn't do the same to a pug, yet that same person might find both to be abominations. As a whole, humans are very speciest. What we find cute, fluffy, or attractive, we'd give more priority to. What we can use, what is big, meaty, productive, we also use. In the UK, all vertebrate animals (and a particular species of octopus) are protected by the Animal Welfare Act (2006). Then you get all into the topic of animal sentience and stuff, which is all above me, because I struggle to get my head around it. But in short, animals we deem as being more sentient tend to get higher priority than those who don't because we can relate to them and because they're more interesting. Conservation charities advertise for "adopt a tiger", yet the money certainly wouldn't just be spent on tiger conservation as there are endangered plant species, too. You'd never see a pitch for "adopt a Myrtle Elbow Orchid" because it just doesn't appeal!
It's again, all very circumstance-based, but we perhaps differentiate our treatment of different animal groups more than we should.
4) Should animals that are dangerous to humans be controlled, by either lethal or non-lethal methods?
Well, lots of things can be dangerous to humans. We keep bulls, which can be dangerous. Pigs will eat you. There is a danger of death with most animal husbandry. There was a small uproar in London about an urban fox getting into someone's home and biting the finger off of their baby, which prompted a "we need to do more to control urban foxes" response from the Government, at the time. More people get bitten by dogs than foxes due to our close proximity to them and the fact that foxes often shy away from human contact. Although a growing population of urban foxes coexist with humans, attacks are so infrequent that the news story was unreasonably disproportioned. (Part of my personal feelings on this is due to certain people maybe wanting to reintroduce the sport hunting of foxes). Dogs frequently do more damage though; do they need to be more controlled than they are currently?
As for large, dangerous animals, they tend to live outside of human contact and if anything, humans do more damage to their species than their species do to ours. If there really is a major problem with say, man-eating animals, then control measures might need to be taken, but only when needed. I feel that control measures should aim to be non-lethal and welfare-friendly, and lethal treatments should be used only when there is no other possible option.