Could there be a planet at the barycenter between two or more stars revolving around each other? In binary star systems, could there be a planet the stars revolve around, with eternal day on all sides?
1st scenario: Imagine a binary system consisting of two Sunlike G5V stars of 1 solar mass each, orbiting each other, and at the barycenter between them is a planet (which would make the planet itself revolve at the same horizontal speed around its axis, making each sun appear always above the same location on the planet). If the stars revolved around the planet both sides would be illuminated similarly. I don't see why it should be impossible. A planet that once was the outermost one in orbit around one of the stars got ejected from its orbit by the other star's gravity, migrating into the barycenter between them.
2nd scenario: Imagine there's a planet at the Lagrangian point between Alpha Centauri A and B. If the stars revolved around the planet both sides would be illuminated similarly. Would that be possible?
Do situations like those above occur or was such even observed?
No.  Such an arrangement is at best &quot;metastable&quot;.  That is, although there are periodic solutions to the three body problem (stable orbits) an infintesimal perturbation (eg the proverbial butterfly flapping its wings) will push the system off the stable orbit and into chaos.  Getting a planet to remain at the barycentre is like trying to balance a pencil on its sharpened point.
With two bodies, each orbits around the barycentre.  But with three bodies, the bodies don't orbit around the three-way barycentre. And a planet placed near the barycentre of two stars will not tend to remain in orbit around that point.
The Lagrangian point L1 is also at best metastable. Satellites that orbit the sun at the Earth-Sun Lagrangian point need to fire their engines and do regular &quot;station keeping&quot; to keep them from drifting away.
The L4 and L5 points can be stable. Bodies at the L4 and 5 points are called &quot;Trojans&quot;.  However no Trojan exoplanets are known.  A Trojan planet would see the two stars separated by (a variable amount averaging at) 60 degrees