"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? "

These are contradictory. If the drainage outside can definitely be
improved, that doesn't sound like a water table issue. I'd lower the
float level so the pump comes on when the water level is still quite
low. See what happens, it's the easiest thing to try. If it works set
very low, you could then try raising it a bit to decrease the cycling.
A second sump pump could help, depending on if the water is showing up
at areas farthest from the current one.

As far as what's going on outside, I'd get out there during a heavy
rain and take a good look. It's the best way to see where water is
running, pooling, etc. You may be surprised at what you find. I'm not
a big fan of dry wells, unless absolutely necessary because there's
nowhere else for the water to go. The first line of defense is to
channel the water away from the house as far as possible. Six feet is
a good start, 10 would be better. Make sure there is good pitch, like
3/4" per foot away from the house and that there is nothing in the way,
like plastic landscape borders, etc.