The light of which star is next to reach the earth? I heard that the sky wouldn't be dark at night if the light of every star had already reached the earth.


Of which (currently invisible) star will the light reach the earth next and when will that be?
What was the last star that we can now see at night to reach the earth and when was that?
How can we know (1) if the light hasn't even reached us yet? Or can't we know and will a star suddenly appear without anyone being able to predict that?
I think you may be thinking of Olbers' paradox
. This supposes that if the universe were infinite in time and extent, and stars were more or less randomly placed, then every line of sight would end at a star, and so the sky would be as bright as the surface of a star.

The paradox can be resolved by supposing that the universe is not infinite in time, but had a beginning.

We don't see stars suddenly appearing as their light reaches us. What we see is stars in all stages of their development. When we look the nebulae where stars are born we see clumps of gas in the process of collapsing to form stars. These are called protostars. Now the process of stellar formation takes a long time: about a million years for a medium sized star. The start of a star is not a sudden "turning on". Instead, as the core of the protostar shrinks and heats up, nuclear fusion starts; slowly at first, then increasing until it is able to prevent further collapse and heating.

Protostars usually are dim in visible light, they can be found with infra-red telescopes.

Since the process is slow and there isn't a sudden turning on, your questions cannot be directly answered. However, there is an object HOPS383 which is 1400 light years distant, which between 2004 and 2008 had a "growth spurt" and became visible in infra-red for the first time. It can't be seen in visible light yet - so in a sense this protostar answers your question 1 and 2, in different wavelengths.