Gravity Variant on a Tether

My questions concerns the gravity gradient on an orbiting tether, let's say
1,000 km long.

The center of mass for the tether would determine its orbital velocity so
the portion of the tether above the c.o.m. would be travelling faster than
its orbital velocity would require and the tether below c.o.m. would be
travelling slower than its orbital velocity would require.

In a microgravity environment, the sensation of freefall is experienced
because the forward orbiting velocity is equal to the downward pull of
gravity.

What happens when you are below the tether and your orbital velocity is
slower than it should be and is thus less than the pull of gravity? Will
freefall not exist in the stations or gondolas below the tether's center of
mass?

Also, above the tether's c.o.m. would gravity be pulling you "upwards" away
from the Earth?

I hope I've gotten the fundamentals correct.

TangoMan