How close together can two stars coexist?

Yes to all of the above. But you can answer most of those questions yourself, simply by considering the interplay between our own earth, sun and moon. Our moon and earth orbit one another as they circle the sun. Once you have thought this through, perhaps you can reword your question. For example your own planet has ‘daylight’ at all times, and pulls it off with but a single star.

Well he meant that the planet doesn’t experience night anywhere.

It has been a long time, but I’m agreeing with Dutchess.

I’m sending this to our resident astronomer.

That’s better. Now for THAT part of the question, how bright is “daylight”. How bright do the stars have to be? How massive, how hot, how far from the planet?

Suppose, for example, that our own solar system was as close to Vega or Sirius as the triple star system now closest to us. We would have perpetual ‘daylight’ big time.

I’m having difficulty visualizing perpetual daylight. The only time that would exist would be a position between the two suns. Only two scenarios could achieve that. One would be if the planet were trapped between the suns. The other would be if one sun were dominant, and the other orbits is, as do the planets, but at a speed which allows it to remain on track with said planet. It also would have to have gravity insufficient to draw planets into orbiting it.

In the middle of a globular cluster like M13 stars are a few light-months away from each other.
Here is a thread on it on Cloudy Nights
https://www.cloudynights.com/topic/32899-about-how-far-apart-are-stars-in-a-cluster/

Perpetual daylight is possible and easily imagined. Suppose two stars orbit one another. Suppose our own yellow dwarf sun was the binary partner of blue giant, with the 2 stars separated by such distance from our planet that the light from the large star matched the intensity of that from our sun. You can have any variety of light shows depending on the the intersection of the star/sun orbit with that of the sun earth orbit. If you suppose the earth, sun and blue giant somehow confined to the same plane (what are the chances?) you can conclude that the amount of light received by the planet overall might be close to constant but just as now, day and night must depend on where you are on the rotating earth as well as the position 2 stars in our sky. It is exactly the same concept as our sun and moon. There will be times when both stars beam down and on the opposite side of the earth it will be completely dark. Now suppose the plane of the earth/sun orbit intersects the star/sun plane at a right angle. What then?

And there are myriads of stars circling one another so closely that they are both physically distorted. There are pairs of stars racing around one another in a matter of hours, some of them so close that one steals material from the other.

1. I don’t understand this question.

2. Planets can exist in binary star systems. The two types of planetary orbit are where a planet orbits one of the stars as its satellite (s-type), or it orbits both stars with a barycenter between the stars as a typical planet would do (p-type).

3. No. This would not be a stable system.

4. No. An s-type orbit would have the planet lit from both sides only when it’s between the stars. It would still get a dark phase and can’t always be lit all around at all times.

Thanks @Kropotkin