Change in Entropy and Enthalpy found over a temperature range and asked to compute the change in Gibb's Energy
This should address your confusion about constant T:

$$G = H - TS \Rightarrow dG = dH -TdS - SdT$$

$$ \Rightarrow \text{(at constant T) } dG = dH -TdS \Rightarrow \Delta G = \Delta H - T \Delta S$$

So, yes, you need $\Delta T = 0$ for $\Delta G = \Delta H - T \Delta S$ to hold.

However, this holds at any T.  

E.g., suppose you do the reaction at $300 K$. $\Delta T = 0$, so $\Delta G = \Delta H - T \Delta S$ holds.

But suppose you repeat reaction at $350 K$. It's still the case that $\Delta T = 0$, so $\Delta G = \Delta H - T \Delta S$ still holds.

As far as the question itself goes, I agree with Chet Miller that it appears to be badly worded.  It seems what your tutor meant to say was:

"Between 300 ºC and 350 ºC, assume that $\Delta H$ and $\Delta S$ are constant, with values of 1700 kcal and 15kcal/K, respectively. Calculate the range of temperatures over which $\Delta G &lt;=0$, i.e., the range the textbook labels as 'spontaneous'".