Generally, I think that both types of questions can be on-topic, with the modification "managing wildlife" instead of the strongly charged "slaughtering wildlife". However, to be on-topic questions must be framed in a biologically meaningful way, for instance by relating to processes, mechanisms and experimental procedures, but this is the case for all questions at the site.
1. Managing wildlife
Pest management is a scientific field in its own right (see e.g. the journals Pest Management Science and Journal of Pest Science), and the same goes for Wildlife management, which includes for instance hunting, human-wildlife interactions and conservation (see e.g. Journal of Wildlife management and European Journal of Wildlife Research). Both fields can also be seen as subfields of applied ecology (sometimes published in more general journals, e.g. Journal of Applied Ecology and Ecological Applications), and to disregard this competely and deem such questions as off-topic does not make any sense for a biological StackExchange. More personally, the research at the department where I'm currently working consists to a large extent of applied ecology, either dealing with pest management, wildlife or conservation biology. Therefore, I think questions regarding pest management (or wildlife management in general, including hunting) can be on-topic if they are framed in an appropriate way, irrespectively of being academic or of general/practical interest (we do not really make this distinction in other topics).
The issue of pest management is also not only dealing with insects in agriculture or forestry. For instance, management of rabbits in australia, bullfrogs, racoons and many rodents come to mind, but there are also many other examples. It is also important to remember that some pest species are introduced and/or cosmopolitan species that are also relevant for conservation biology, since they put pressure on rare indigenous species. Therefore, management of wildlife or pest species is not simply a black & white issue of "bad" human-centered management and "good" conservation biology.
2. Animal experimental procedures
Animal experimentation is used as a method in medical and biological research, so I do not see any reason to ban such questions. We allow all other types of questions that deal with biological research, so why should these be excluded? Generally, these methods are also legal in many/most countries in different forms, so to ban them would be strange from this perspective as well.
Regarding both topics, I can also think of questions that I would dislike strongly from a personal/ethical standpoint. However, I suspect that many of these questions can be closed for other reasons as well (too broad, poor background, opinion-based), and we should be able to deal with them on a case-by-case basis. For instance (using the example from @WYSIWYG), I struggle to see how a question in "favour of killing of endangered species" can be framed in a reasonable and biologically interesting way. I also think that @resonating has a good point in that the community can choose "to exercise its moral compass", and users can downvote and choose not to answer questions that they dislike from an ethical perspective. At the moment, I do not think that a custom close reason is needed to deal with these issues, since we have very few of these types of questions and I also think it will be tricky to formulate such a close reason that isn't subjective and/or doesn't exclude topics that should be on-topic for the site.
If we start evaluating the ethics of individual questions on hunting, pest control and animal experimentation, we are on a slippery slope towards moderation that could easily turn subjective and inconsistent. There are many other topics that can be just as controversial as the ones mentioned above, for instance cloning and genetic modification. It would be extremely unfortunate if go in the direction of letting the ethical qualms of individual users guide the entire community. Also, as mentioned above, I think that most "unethical" questions will be closeable for other reasons as well.
In the same vein, I think the suggestion by @GoodGravy that "...questions involving extermination, hunting, or other experimentation that induce pain should require ethical justification" is misguided and impractical. If the ethical justification is to be evaluated, how can this be done in a fair and predictible way? To me, the evaluation would depend on the specific group of users that vote to keep open or close on each individual question. All close-votes natually has an element of this, but the effect is compounded if the evaluation is based on subjective elements such as the ethics of the questions.
Some of the comments and answers mention that only academic advice should be allowed as on-topic. I think that such a division between academic and non-academic is really hard to define, justify and uphold.
First of all, we get a large number of questions that are curiosity based with very little previous reseach from the poster. These can hardly be labelled "academic", even if the question mentions a couple of subjects or buzzwords. To start targetting "non-academic" questions would therefore change the site practices quite a bit.
Second, I don't think you can make a clear division between questions asking for advice and those that don't. To a large extent, this just depends on the exact wording of the question, and questions searching for advice can often easily be reworded in a more general form (the same goes for no-effort homework questions).
Third, I think we should encourage, not discourage, "non-academic advice" questions from professional working in fields related to biology, e.g. fisheries, agriculture, horticulture and forestry. They could be a great addition to the site if phrased in terms of relevant processes while clearly explaining the problem that has to be solved. In fact, if I would have to choose between one or the other, I would much rather have practically relevant, applied questions in these fields that requires answers based on biology than many of the top voted questions at BioSE. Brief examples of interesting applied question could be:
I'm trying to manage a population of wild boars, and am wondering how to best allocate the hunting effort. Should we focus on hunting the yearlings or older individuals that are reproducing? Wild boars live in tight social groups were only a few individuals reproduce. To target the dominant reproductive individuals therefore seems sensible, since it will drastically reduce the reproduction in groups. However, I've heard that such hunting practices can result in the groups splitting into several smaller groups each with reproductivly active individuals, which could lead to higher reproductive rates in the long run.
Without knowing this area particularly well, I know that such consideration has been an active area of research. Answers can be based on the social dynamics as well as the sensitivity of the population growth rate to changes in different vital rates (i.e. the same method that is often used in conservation biology, but from the opposite perspective).
I've read that biological pest control using natural predators can be an effective way to manage pests. I'm now considering using this more actively in my own wheat fields, by using wider unmanaged field borders or unmanaged plots within fields to support larger predator populations. However, to be able to weigh the pros (increased pest control) and cons (smaller planted fields) I need to know more about how large the added pest control really is (the effect size). What has been observed in previous studies? My fields lie in the northern part of Germany, and it is also worth mentioning that my "field margins" are relatively wide due to relatively small individual fields and wide ditches. Is it likely that I will observe a postive effect from additional measures to support biological pest control?
Answers to this question could be based on the large body of literature that has tried to quantify these types of effects in different types of landscapes, field geometries, crops etc, while also trying to weight the positive effect against the less effective management of more fragmented fields and smaller planted areas. Also note that both these examples could easily be posted by users from a non-academic background and perspective.
Overall, I think that the premise of the question as well as several of the comments and answers are somewhat ignorant, especially to the fact that pest management is a huge established field of science, which most users of the site probably depend on indirectly for the food that is found on their plates.