What is the recommended temperature for cooking eggs?

The FDA recommends 160°F (71°C) for dishes containing eggs. If you're scrambling eggs or making an omelette, as long as they're coagulated/solidified, they'll be safe.

The easiest way to confirm this is with sous vide eggs. For example, this Food Lab sous vide egg guide has 155°F, 160°F, and 165°F eggs:

which clearly show that the white won't be totally solidified until 160°F or even 165°F, even when cooked for an extended period of time.

(To be clear, those lower-temperature eggs are actually safe if cooked for sufficient time. The point here is just to show coagulation versus temperature, to demonstrate that when eggs cooking in a pan, you can use the solidity as a temperature indicator.)

If safety is really your primary concern, boil the eggs long enough until the yolk starts to go blue, which should take around 10-12 minutes for a medium sized egg. They're pretty much as harmless as they can get then.

Other than that, any non-solid particle should technically considered to be a health hazard, if you use "health hazard" in the most liberal sense of the word.

Eggs can be consumed raw if you want, there's just a bigger chance anything bad in there might actually affect you. Note that "something bad present" does not equal "you'll get sick". It might make you sick, but merely presence does not equate getting sick. As you reduce cooking time, chances increase that some bacteria was still left alive, and thus your chances of consuming that bacteria increase, and thus by the power of numbers, your chances of you being affected increase.

In general, wait at least until the whites start to solidify, and you'll be good.