What is the recommended temperature for cooking eggs? Omelette, scrambled eggs etc. Are there any visual cues by which you could know if the eggs are "Safely cooked"?
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.)