Abandoned Hall

No. Abandoned Hall's "when a player plays a spell" condition is fulfilled only when that spell finishes resolving.^420.4.a A countered spell does not resolve and is not considered played for triggered abilities, so Abandoned Hall's ability is not put on the chain.^420.4.a.1

After the spell resolves, Abandoned Hall can choose only a unit at Abandoned Hall controlled by the player who played that spell.^{190.6.c355.9.b355.10}

• Moved into Abandoned Hall: The unit can be chosen if the spell's player controls the unit when Abandoned Hall's triggered ability finalizes.
• Moved away from Abandoned Hall: The unit cannot be chosen, even if the spell's player controls the unit, because the unit is no longer at Abandoned Hall.

Abandoned Hall's trigger condition is not fulfilled until the spell has finished resolving.^420.4.a The resulting ability is then added as a pending item.^383.3402.1 Abandoned Hall's triggered ability chooses its target when the ability finalizes.^403.2

Yes, if the player who played the spell controls that unit when the ability finalizes. The spell starts playing the unit during its own resolution, so the unit waits on the chain as a pending item.^354.2354.3 When the spell finishes resolving, Abandoned Hall's triggered ability is added as a newer pending item.^{420.4.a383.3402.1} The unit finalizes before that newer ability because pending items finalize in append order.^337.1.b It immediately resolves and enters Abandoned Hall.^337.2359.2.c When the triggered ability later finalizes, it can select the unit played by the spell as its target.^{190.6.c355.9.b355.10403.2}