So, I finally managed to solve it with a workaround.
Rather than using an Object or Actor Reference, I tried using a Class Reference to DamageZone, and it finally allowed me to set an instance for this Class Reference. Then I used Get Class Actor to access “is in damage” and the code finally worked properly as I wanted it to work.
The fun part is after defining a Class Reference when I defined an additional Object Reference it worked too, allowing me to choose an instance for the object. But as soon as I remove Class Reference, Object Reference loses the instance. Still can’t figure out why is this happening (as far as my almost zero knowledge of UE goes) but I can temporarily move forward I think.
Also, your point on placing the overlapping process on Character side is absolutely correct. I need to change it.