Does moving a unit using Flash exhaust it?

No — Flash just moves the units; it doesn't exhaust them. Exhausting is only the cost of the Standard Move (Rule 144.2). Flash's costs are its 2 Energy (plus Reaction timing), nothing else, so units moved by it stay in whatever ready/exhausted state they were already in, and a ready unit moved by Flash is still ready afterward.

That means Flash can reposition a unit and leave it available to act — e.g. you can Flash a unit to base and it can still make a Standard Move later that turn if it was ready. It also picks up already-exhausted units fine, since the effect doesn't care about their state.