This is due to the fact that they work slightly differently from hero cards most notably, they set the hero's Health to a predetermined amount rather than retaining the hero's current Health.
However, they can still appear in Arena games through random effects.
Experience and win count rewarded at the end of the match will also apply to the new class. In this case, the class the hero card belongs to will replace the player's original class for the purposes of mechanics such as Discover. For example, a Hunter who plays Deathstalker Rexxar will always transform into Deathstalker Rexxar, regardless of whether they are using Rexxar or Alleria Windrunner as their hero, and a Priest can transform into another class' hero card if they obtain that card from Thoughtsteal or Mind Vision.