> I am stuck with figuring out what to do with the existential quantifier in the third premise. 

Let us have a quick look.  You are aiming to derive `Gse` from the premises, so first use universal elimination on the third, and then...

     | Pd ⟷ (Hj & Mj)
     | Gsd
     | ∀x∀y∃z(((Gxy & (Py ➝ Pz)) & Rxyz) ➝ Gxz)
     | Pe ⟷ ∀x(Hx ➝ Mx)
     | Rsde 
     |- 
     | ∀y∃z(((Gsy & (Py ➝ Pz)) & Rsyz) ➝ Gsz)
     | ∃z(((Gsd & (Pd ➝ Pz)) & Rsdz) ➝ Gsz)


… Nope, you cannot do anything else.  You cannot establish that term `e` is a witness for that existential, and further you cannot derive that `Pd ➝ Pe` .  You have no route to deriving `Gse` ; it simply is not entailed by these premises.