I have a sump bump in the basement that generally only operates when
we get a lot of rain.

When we get particularly heavy rains (most recently 2" in 24 hours) I
get water in different parts of the basement even though the sump is
pumping regularly. The water appears to be coming up from the floor.

The top of the sump pit appears to be at the same level of the
basement floor. The float is set very high, so that the pump does not
turn on until the pit is almost completely full (less than 1" from the
top). Would setting the float a bit lower cause the pump to get rid of
the water before it rises to the level of the floor? During heavy
rains the pump runs about every 5 minutes.

Maybe one pump is inadequate and I need another one in a different
part of the basement? Or maybe it is not deep enough?

The drainage outside the house can definitely be improved. The ground
is sloped away from the house and the gutters extend around 6 feet,
but I have a feeling that roof water is being deposited too close to
the house. How much of an improvement can I expect by improving this
situation (e.g. using a dry well or french drain) since the issue
appears related to the height of the water table?

I do plan to get a professional to check everything and give advice,
but I really want to learn as much as I can first. Any help would be
greatly appreciated.

--
Dave