Based on my reading of the explanation text for free move and encumber, a free move should not ignore encumber effects. (I vaguely recall that other free move cards, like scamper, don't ignore encumber effects, but I could easily be remembering incorrectly... anyone know?) This is either (1) confusing mechanic text, if free move should indeed ignore encumber effects, or (2) a card implementation bug, if elvish mobility shouldn't ignore encumber effects. For (1), I would suggest that the free move explanation text read: "You are not required to stop when next to an enemy, or into difficult terrain. Ignore encumber effects and any other hindrance to movement except occupied and impassable terrain.". For (2), I can see a bunch of factors that could contribute to a card bug: two characters with the same name ("Tineva"), one encumbered one not; encumbered character was under the control of different player than the one who played "Elvish Mobility"... Debug info: "BATTLE LOG: Scenario=MP Dungeon,Room=neoncat vs SgtSteve,RoomID=247187"
Don't think it should with the current wording (only push and slides should), even though I remember suggesting free move should be immune to encumber when the team move card and encumber got changed. My guess it the card just checks the caster for encumber effects...
As far as I know, encumber only subtracts movement when the afflicted character attempts to move itself. Not when he moves another, and not when somebody tries to move him. Whether that is intentional or not, i'm not sure of.
Yes, Encumber only triggers when the move is instigated by yourself. So, Push cards aren't affected by Encumber. Elvish Mobility is being treated as a Push but it has a special rule to make sure that if an enemy Elf is allowed to move, the owner of that Elf controls the move. So it's slipping through the Encumber net. A weird side effect is that the Elf which plays the Elvish Mobility is the only one that is affected by Encumber! I think it would be clearer if all moves triggered by this card are Encumbered.