Is an aquifer ever likely to get saturated? I have no expertise in this area, but in the US at least the concern is always over aquifers being depleted, not so fully saturated that they can't absorb more water. My understanding was that it takes very long time for water to actually get from ground level down into an aquifer.

At some point the ground cannot accept anymore water, no matter what.

Proof: oceans exist.

The key to the whole thing is if you can slow it down enough so that a sudden deluge over a day/week/month has time to slowly drain out.

Getting water into an aquifer takes time depending on how the soil is, but eventually the aquifer would rise to the surface and you'd have a marsh. Or it would rise higher and you'd have a lake.

This depends entirely on where you live. Sure in the Southwest, there's a big problem with drought. But along the Gulf Coast, the water table is astonishingly close to the surface. When my family put in an in-ground pool in Louisiana, they had to start filling the concrete pool with water even before finishing it because if they left it empty, the surrounding water table would push it up out of the ground like a floating boat.