Unusual voltage drop at BJT (2N2222)

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The model of each transistor is not dependent on what it will be connected.

Yes it is possible, but you must have into account the sign of the currents and which voltage controls the collector source current, Vbe or Veb.

You can take the NPN model and apply Vbe = -Veb (that's by definition). Then, on the equations for the NPN model you'll have ib = -Veb/rb, ic = -gm Veb and ie = ib + ie;

But now all currents are negative and the arrows pointing against the flow. Since that's not intuitive and we're free to choose the signs, we consider positive the currents that flow outwards and turn around the current source. That translates to the equations to multiply the right members of the equations by -1. This way, you have the PNP model.

As you see, It's a matter of avoiding minus signs. Just like in large signal analysis we say VEB = 0.6V instead VBE = -0.6V for a PNP transistor.

Unless your data is very precise, and includes values very close to VCE=0, you won't have enough information to determine VBE precisely.

Generally the VBE will be around 0.65 V for most transistors while operated at 'normal' collector currents. If you operated at 1/100 of this value, VBE would only change by about 0.12 V. If you operated at 50 C hotter than room temperature, the VBE would also only change by less than 100 mV. So if you don't need a value more precise than that, I'd suggest using 0.65 V as a good estimate.