How much oil is necessary to fry/cook eggs?

Technically, an egg is not "fried" unless there is at least some oil involved. So even though you could cook an egg in a very well-seasoned cast iron pan with no oil, it wouldn't technically be a fried egg.

The primary reason you use oil, though, is to keep the eggs from sticking. So in a pan like yours -- a worn-out nonstick pan that's not really nonstick -- you're going to need oil, and probably a fair amount.

The exact amount is going to vary according to three factors:

1. How large your pan is
2. How many eggs you are making
3. How "sticky" the pan is

So if that nonstick pan is only 6"/15cm, just a tiny bit scratched, and you're making two eggs, you can use just 2 tsp of oil. But if you're cooking 8-10 eggs in a 14"/35cm nonstick pan that's completely scratched up, then you'd need more like 4oz/100ml of oil.

Which brings us to the other reason to use oil: flavor and texture. Some ways of making fried eggs use even more oil. For example, the standard Thai fried egg is cooked in a pool of oil 1/2"/1cm deep or more, in order to get lacy, crispy whites.

I'd say not more than a table spoon (around 15ml), maybe a little more if you feel your eggs are sticking; you need to experiment, but I think you should use as little as possible.

Use oil (or other fat) to help crisp up the egg.

Butter will work better for scrambled eggs (IMO).

The amount of oil required is dependant on the result you want.

At minimum, you can fry an egg in a teaspoon of oil, but if you want the top basted you will need sufficient that you can splash some over the top to finish it. Otherwise you either have to fry it for longer, or risk the white still being runny. Alternatively, abandon the sunny-side-up idea & flip it.

Butter will burn unless you keep the heat down, but it's a viable alternative if you have the patience. It's not what I would consider a 'true' fried egg, though. Hot oil, fast cook is my ideal. Slight crisping around the edges.
As mentioned in comments - if you have bacon fat, that's the best.

Catering fried eggs are a different thing altogether - to save having to carefully monitor each egg's progress, they're cooked in maybe an inch of oil, so they float.
I wouldn't suggest this as a particularly delicious method for home cooking, they come out rather pale & insipid. Sometimes they'll grill them to finish.
Just to cover all bases - some caterers will fry eggs on a griddle/hotplate surface. This is similar to the pan method, but you can't baste.

Scrambled eggs are a whole different thing - though you would traditionally start with a butter coating on the pan, for flavour, the result is not 'fried', it's merely 'heated & stirred until it mostly solidifies'. If you have a really good non-stick pan, you can make scrambled with no oil/fat.

Fried, on the other hand, really can't properly be called fried unless it's in oil/fat.
That's the very definition of flrying.

Although the accepted answer suggested around 15 ml, I would do around 5 ml (a teaspoon) for a single egg. Just make sure you don't unnecessarily coat the whole pan, and you should be good with a teaspoon.

For me, pans are remarkably non-stick right after they have been used (still hot), meaning I can fry an egg with the super thin layer of oil left on the pan.

Under what circumstances should I use oil for cooking eggs

I will turn that around: the circumstance where you will not use oil is when you have a working non-stick pan (so not the worn-out one you described) and you prefer eggs made without oil. If any of these conditions is not true, you should use oil.

and how much

The range is huge. Upwards, it is basically unlimited - I haven't had deep-fried eggs, but I don't doubt that somebody is making them. What is more interesting is the lower limit. You have to have a layer of oil that is sufficiently thick for the egg to be able to slide on it without touching the pan, no matter how many milliliters it takes This is quite thin, less than a millimeter of depth is sufficient, although you might not be able to get a continuous layer that thin on a failing nonstick pan.

Make sure you are not using less oil than that. If you have a normal pan and use too little oil, the eggs will stick. If you have a nonstick pan, the oil will polymerize and gum the pan up.

The heat matters as much, if not more, than the oil, unless you're basically shallow frying (like, enough oil to not cover the egg, but still get halfway up it or so). Get the heat right, and only a bare touch of oil will be enough (or maybe, none at all). Get it wrong, and even a fair amount will still see you sticking.

What's really important is that it's hot enough before putting the egg in. Oil will help you here, because it will transfer the heat a bit better than the pan itself will. Make sure that the pan is preheated (meaning, as hot as you want to cook the egg on), and that the oil is also preheated (so, on the pan for at least 15 or so seconds before the egg is).

Also make sure you're using the right level of heat; if you have a laser thermometer or an electric griddle/induction range, this is easiest. Aim for something like 325°F/165°C. Much higher and your oil might burn, much lower and you'll end up sticking.

Ultimately I use just a tiny bit of oil - a faint spray from an oil mister - and don't usually stick when I cook my eggs, on a non-teflon nonstick pan (so, a pan that isn't quite as non-stick as the teflon ones).

One other note: you can season non-stick pans just like you season cast iron! Put a thin coat of oil on them, then heat them up until the oil is quite hot (some smoking is okay, but it will discolor the pan if it's not a dark color pan). If you're using Teflon pans (PTFE), be careful not to heat the pan beyond the safe temperature your manufacturer recommends, but the hotter the better for this up to that point. Let it go for a while, at least a minute or so. Then turn it off, and let it cool completely. Finally, wipe it out with a towel (but no soap/water). This page for example goes over this. You can do this to an older pan and regain much of the nonstick ability, as long as not too much has actually flaked off!