In project #3, if I computed the output for given x(t0), y(t0), and got the correct base and exponent in t1 when done(t1)=1, and then I keep the inputs unchanged - should done(t)=1 for all t>t1?

After every computation is complete (i.e. once \mathit{done}(t) = 1) the circuit starts a new computation with the appropriate inputs. In your example, if you never change the inputs x and y, once the first computation finishes, the second computation is performed “from scratch”.