Help understanding this equivalent mechanical circuit of a piano string?
What do the hollow circles at the top and bottom portion of the circuit represent?  


Those circles are the connections of the different elements indicated by the curly brackets below the schematic. Each element is represented by an electrical equivalent schematic and these circles show the border/connection of each schematical representation.
They also show the points/nodes across which the across variable is being measured (see below).


  Which way does the circuit flow throughout the entire diagram? Can you add some arrows to illustrate?  


I think you mean something else. The circuit doesn't flow.
Since each \$f_x\$ is shown with an arrow, in this representation, the force is the through variable  (analogue to electrical current) and the vibration velocity is the across variable (analogue to voltage).(1)

Nota bene
An arrow defines the sign of the variable, but does not dictate the direction of the current!
So, force \$f_1\$ can flow into the soundboard and into the transformer towards the "horizontal string".


  [..] Is the whole bottom of the circuit in Fig 3 ground?  


Yes.
I would conclude the same based on Fig 2. The masses M of both strings (shown as capacitors) share the same ground connection in Fig 2. So, their shared connection (the bottom connection in Fig 3) should therefore also be ground.
Next, \$u_1\$ and \$u_2\$ are across variables, where \$u_1\$ and \$u_2\$ actually are voltage potentials with respect to something. Equation (3) in OP's refered paper only holds when these voltage potentials are with respect to a zero voltage reference, or, ground.

Combining points 3 and 4:


  eg. The arrows for force (f) and velocity (u) indicate that a horizontal string excitation (f3,u2) will pass through the bridge pin transformer and create an excitation (f2,u1) that can then pass through the soundboard and this should lead to ground as per Fig 2.
  eg. How would a horizontal string excitation make its way into exciting a reactive vertical string excitation in this model? Or vice versa? They should be able to excite one another directly in either direction through the bridge pin. Where does the signal flow both ways in Fig 3 to allow this?


The statement "a horizontal string excitation (f3,u2) will pass through the bridge pin transformer" is correct when you consider (f3,u2) being energy (or: an energy pair).
So, "a horizontal string excitation energy (f3,u2) will pass through the bridge pin transformer" and will result in an excitation energy (f2,u1) which is seen by the soundboard as well as the vertical string.
The same way, a vertical string excitation energy is seen by the soundboard as well as the transformer and therefore by the "horizontal string".



(1) Other analogies define force as effort variable and velocity as flow variable.