Do two objects (two large hunks of lead suspended from two strong strings) exert a gravitational force on each other?

Yes

Explanation:

Two objects weather they are suspended from two strings or not, do exert a force on each other known as gravitational force.

It is even explained in Gravitational law which states two objects do exert a gravitational  force on each other which is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Though these forces are very small as their masses is not that much thus an equilibrium state is formed where objects do no move but they do exert gravitational force on each other.