Does moving a unit begin a chain? Since a move does not open a chain, but units take the "standard move" action to move and exhaust themselves, and card actions open a chain.

No — moving a unit never starts a chain. The Standard Move is an inherent ability and a discretionary action, not an activated ability: the cost (exhausting the unit) doesn't put it on the chain, and the move resolves instantly with no reaction window (Rule 446.3.c).

Two exceptions create reaction windows, but not from the move itself: if the unit has a "when I move" trigger, that trigger goes on the chain (and opponents can react to it), and moving to a battlefield you don't control stages a Showdown, which opens focus/action windows. A move to a battlefield you already control does neither — it just happens.