Assuming that I've already cooked the poultry (or eggs) to recommended temperatures - does it need to be refrigerated immediately, or is it safe to eat if it's been left at room temperature overnight?
Time for a safety lesson.
When you are dealing with food products that can spoil, what you are concerned about is the Danger Zone, which is the temperature range at which bacteria multiplies fastest. While specific numbers will vary, a decent rule of thumb is 4-60 degrees Celsius. What that means is any food prone to spoilage must be kept out of that temperature range, and if it is going to be within that temperature range, it must be for less than four hours, aggregated across the product's lifetime. What this also means is that when you are chilling hot food you need to do so as fast as possible, in order to minimize the time spent in the Danger Zone.
Foods that are dangerous are the ones you would expect: anything which requires refrigeration. Potatoes, for example, are perfectly safe to keep at room temperature when raw, but must be refrigerated once cooked. (Perhaps a poor example, as potatoes fare better when kept below room temp but above fridge temp, but you catch my drift I hope). So we're talking about raw meats, most dairy (very hard cheeses may be kept at room temp with no ill effects other than possible softening), many vegetables and fruits, that sort of thing. Eggs and butter can be considered a somewhat special case; eggs can be kept at room temp with few problems (and in fact eggs used to be kept right out on kitchen counters, in the shell of course, not after they have been cracked), while butter can generally be kept at room temperature for a few days before the fats will start to become rancid (anecdata: we never kept butter in the fridge when I was a kid, except for butter my mother needed for baking, where the temperature is a matter of physics in the recipe, and not health. Never had any problems.) Harder vegetables--potatoes, carrots, parsnips--can generally be kept at room temp without problems, as can many fruits--apples, oranges (all thick-rind citrus really), bananas, and so on.
So. According to the food safety experts, you absolutely should not leave even cooked poultry out overnight, as any bacterial contamination will be severe after a few hours. In real terms, unless you are very young/old or immunocompromised in some way (e.g. leukemia, chemotherapy, HIV/AIDS, etc) you're probably going to be okay. I am a professional chef, and at home I'm pretty lackadaisical about expiry dates, how long most things have been sitting out, etc. In the context of work, however, I am extremely anal about rapid chilling/heating and keeping food safe, and I will always advise other people to be cautious with their food; what I choose to do with my own body is my concern. When In Doubt, Throw It Out is always good advice; there's a reason it was stencilled across the inside of the walk-in refrigerator door at one restaurant I worked at.
You should also bear in mind that much of the time bacterial contamination is not detectable by the eye, nose, or tongue until it has become severe. Even when contamination is not detectable with our senses, serious illness can occur (see, for example, recent outbreaks of E. coli, listeria, and salmonella, none of which can be detected with eyes/nose/tongue). So the 'smell test,' while in use by almost everyone who cooks food, is really not all that reliable as a gauge for spoilage; taste and smell can definitively say that a given product has gone bad, but it cannot say that a given product is definitively okay to eat.
(As a side note on food safety, if cheese has gone mouldy, don't trim off the mouldy bits. Just throw it out. While much of the time any such mould will be entirely harmless, it's not worth taking the chance over, especially given that if mold has bloomed on the surface of the cheese, spores will be present on the entire surface, with penetration into the cheese itself. Seriously, just toss it, and re-examine how you store your cheese and/or how much you're buying at a time).
Even easier to remember: keep hot food hot (above 60C) and cold food cold (below 4C). Make things hot as fast as possible, and cold as fast as possible.
(Note that I am a cooking professional, not a medical professional, and nothing here should be construed as medical advice. Always err on the side of caution.)