Eberron campaign setting page 23 says warforged healing is halved, but it does not specify if it's rounded up or down, so I don't know if lesser vigor, for example, would heal 1 or 0 damage per turn.
What is the rule for this?
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Actual healing is halved, rounded down (per the 'always round down unless otherwise stated' general rule).
Fast healing is an ability that constructs can benefit fully from:
Cannot heal damage on their own, but often can be repaired by exposing them to a certain kind of effect (see the creature’s description for details) or through the use of the Craft Construct feat. A construct with the fast healing special quality still benefits from that quality.
(emphasis mine) So presumably living constructs can as well, and so should heal the full amount from spells that grant fast healing.