Suggest possible structures for the compound PF3(Ph)(NMe2) using the NMR data
I agree that the trigonal bipyramid structure is what we are dealing with, so let me focus on that structure.  

The nmr data shows that 2 of the fluorines are equivalent and one is different. This results in 3 possible isomers as shown below.



Both $\ce{^31P}$ and $\ce{^19F}$ are spin 1/2 nuclei, so each of these isomers would produce phosphorous and fluorine nmr spectra with splitting patterns and integration ratios consistent with the reported data. That is, the phosphorous will split each different fluorine into a doublet, the two equivalent fluorines will split the singular  fluorine's doublet lines further into triplets.  The one non-equivalent fluorine will split the other two equivalent fluorine doublet lines into doublets.  Similar analysis of the phosphorous signal leads to the expectation of a doublet (from the singular fluorine) of triplets (from the two equivalent fluorines).

Two additional pieces of information:


Fluorine prefers the axial position in pentacoordinate phosphorous
(see here)
Because the orbitals involved in axial and equatorial bonding in pentacoordinate phosphorous compounds are different, the axial and equatorial coupling constants will be different.  Generally, $\ce{J_{PF(eq)}}$ > $\ce{J_{PF(ax)}}$ because there is increased s-orbital density in the equatorial bond (see here, for example)


The larger reported P-F coupling constant (959 Hz) is associated with the intensity = 1 fluorine signal.  This suggests that structure I, with one equatorial fluorine (with the larger $\ce{P-F}$ coupling constant) and 2 axial fluorines (with the smaller coupling constant and as preferred electronically - point 1 above) is the structure being observed.

Part B Experiments:

The trigonal bipyramid structure could be confirmed by heating the sample while in the nmr spectrometer.  As Berry pseudorotation commences the 2 fluorine signals should coalesce and determination of a barrier height around 35-40 kcal/m would be expected.  

Also, compounds II and III must have similar energies and both would be expected to be present at equilibrium.  If we heated our sample so that pseudorotation took place, and then upon cooling we saw an additional set of signals, that would be consistent with II or III being our starting compound.  No new signals upon cooling would support structure I.