Assumptions made in calculating melting point from thermodynamic data I'm fine with doing these questions, such as:

For ammonia, enthalpy of fusion is 5.65 kJ/mol, and entropy of fusion is 28.9 J/K.mol

Hence find an approximate melting point:
Well $\Delta G = \Delta H - T\Delta S = 5.65*10^3 - T(28.9)=0$ Which yields $T=196K$. This is quite close to Google's value.

What I'm thinking, though, is that we shouldn't be able (in real life) to determine a melting point so easily, because $\Delta_r H$ and $\Delta_r S$ both vary with temperature, so in reality this isn't as simple as solving a linear equation for T?

I'd just like some confirmation here, as my textbook isn't giving much away on this point :)
In real life the melting point is determined by measuring the temperature. The entropy of fusion is usually determined by dividing the enthalpy of fusion by the melting temperature.

$$\Delta S_f = \Delta H_f / T_f$$

Your text book calculation just solves this equation for $T_f$ to recover the temperature that probably had been measured before.

Note that for constant pressure the temperature is constant $T_f$ as long as the phase transition takes place.