I can see this scene right before my eyes...
Why this sentece is without articles "egy"?
Because "egy" can be omitted.
What is wrong with: "Ez asztal van, nem szék"?
Because "van" must be omitted here. The sentence is just explaining what the object is
Could I use "Ez a" instead of just "Ez"?
No, because then you'd be saying, 'this is the (table)'.
Why can't it be tablazat
Why would the part "nem ez szék" be considered wrong? Does the inclusion of "Ez" at the beginning automatically make all other objects out to be "a" instead of "the", unless specified otherwise?