is there such a doctrine in the easter Orthodox church?
"Levels of Heaven" is an ancient Hellenistic concept, which presupposes a geocentric model of the universe. It posits that the various celestial objects in the sky are somehow part of a series of concentric spheres, which rotate and thus give motion to the planets and stars. Each "level" of heaven is the space between two given spheres, or perhaps the spheres themselves.
So when St. Paul uses the expression, "caught up to the third heaven", I would assume he is speaking of the perceived height he was caught up to, rather than relaying his travels through a "spiritual realm" named the Third Heaven.
Some popular Neo-Platonic religions (evangelicals in America tend to wash them all as 'the Gnostics' due to a lack of philosophical/historical context) believed that one passed through the celestial spheres in order to henotically merge with their monadic singularity god.
Many Christian and quasi-Christian groups today, especially those who practice forms of angel worship, believe that Paul's mention of the Third Heaven dogmatizes the existence of a multi-tiered, mappable celestial realm.
None of this has any doctrinal basis in Orthodox Christianity, except perhaps as fodder for old-timey didactic imagery (I.E. heaven/God and the trials and tribulations of the fallen cosmos represented by the various spheres), or loving speculation on what the Angelic life is like.
Orthodox sometimes refer to those who have fallen asleep as being in Heaven, I.E. that the reposed are with God. Although we believe in Angels and an Angelic "realm", we do no believe that it is some mappable plane of existence with levels and whatnot corresponding to some spiritualized Ptolemaic model of the cosmos.
We also confess the Resurrection of the Dead and the life of the Coming Age. We do not believe that we spend eternity as disembodied minds or that the Coming Age is far off in a place called "heaven". That is paganism.