Honey should be stored at 50-70 Degrees Fahrenheit
Honey is similar in to olive oil and should be storaged between 50-70 Degrees Fahrenheit according to Max Shrem from Slashfood:
Similar to olive oil, honey should be stored at a cool temperature
between 50 and 70 degrees Fahrenheit. So, it's best to store it away
from your oven or stove. Also, it should not be refrigerated. Extreme
changes in temperature will spoil the flavor of honey.
Looks like this confirms that you may want to keep it in a cool place and away from hot temperature where it would cause honey to degrade in quality.
Shrem from Slashfood continues:
Remember
Do not store honey in a refrigerator or near a stove or oven.
Store it in an air-tight container.
Thus, choosing an appropriate container is an important part of the equation.
Penn State has a great article germane to this topic.
Refrigerating Honey
In the article, it talks mentions the crystallizing state of honey if it is refrigerated in no way it harms the honey at all.
Restoring honey from a crystalized form
I know this is not related to your question. But something you asked in the comments. Apparently, honey can be restored from it's crystalized form: placing it in the microwave for 20 seconds can restore it from a crystallized form.
Therefore, you can store it in the refrigerator and heat it afterwards. The downside is that if stored in the fridge too long it'll turn hard and sugary.
Proper Storage Container
Since honey has very little moisture in it it is difficult for microorganisms to grow. Thus, an airtight container is of utmost importance.
If moisture gets in the container it can be an ideal environment for mold and yeast to grow. Honey should be thrown out if it foams and smells like alcohol
Long Term Storage of Honey
Honey can be stored for long periods of time without any issue. Over time, honey will darken and intensify with flavor.
Peanut butters that are not "all natural" include cheaper oils along with sugar and emulsifiers to keep the mixture from separating and to make it lighter and smoother.
That lack of emulsifiers could make a huge difference but it depends a lot on the recipe.
In a normal cookie dough fat is creamed with sugar and eggs are beaten in one at a time which adds a ton of emulsification power from the lecithin in the yolks.
Additionally- the fat will bind with the flour and be baked into the cookies. Like Jefromi- I have not had any problem with the fat from natural peanut butter separating out after baking.
If you made a cookie dough and either left out the yolks or didn't beat it well enough to properly emulsify- and then let the batter sit out for a while before baking I would fully expect the peanut oil to separate.
Best Answer
Per Pick your own (emphasis added):
I could not find a single government or university sponsored publication that discussed how to can homemade peanut butter. If you stumble upon a certain link that does not eschew this practice, and recommends boiling temperature canning, consider them fraudulent.
According to Does it go bad:
Note that all of the even semi-credible sources I could find indicate refrigeration is required.
At ambient temperature, especially in summer, it is going to have a much shorter shelf life, but I could not find any authoritative information. You should probably prepare it in small batches that you can eat within a week, maybe two at the maximum (in winter) if it will be stored without refrigeration.
You also want to store the peanut butter so that the oil (which will probably separate) on top will protect the peanut solids, since it is more difficult for microfauna to gain a foothold in the oil. Keep an air tight lid on it whenever you are not using the contents of the jar, to oxygen contact, and thus hold of rancidity.
Homemade peanut butter will have mostly unsaturated fats (one reason the oil in your peanut butter is liquid at room premature), which are susceptible to rancidity. You will want to discard the peanut butter if it develops rancid flavors as they are highly unpleasant. If you see signs of mold or bubbles or sour or off smells, discard it immediately.
I suggest contacting your local universities, who may have programs or services to provide this type of information for your area and constraints (which I infer includes no refrigeration).
To the enumerated questions: