If the stock prices falls "gently" and the option remains in the money, your delta will converge to 1 and you will have to buy stocks: the gains on your long stock positions will be lower, but the payoff of the option you wrote will be lower too.
If the stock prices falls more sharply and gets closer to the strike, there are higher chances that the option ...
You would be over hedged in your call position if it was delta neutral before the stock cratered. Since you are long delta on the call, you would have shorted stock to make the original position delta neutral.
When the stock fell, your delta would have fallen, and you would buy to cover some of your short stock hedge. However, being long the call, you ...
You are long a vanilla option, so long gamma (positive gamma). If the stock price decreases, so does the delta of your option.
Since you short-sold the stock to hedge, you now have short-sold too much since delta has decreased. As a consequence, you must buy back some stock.
With a long time to maturity, your options have a low theta because their time value decays quite slowly. If there are many months to go, the passage of one day does not change the exercise probabilities too much, whereas short life options with only a few days left have a much higher time value decay. Hence, the larger the time to maturity, the lower theta.
I am not sure what you are trying to do, but I think you are trying to use the Modified Euler Method to find the option value.
If the Delta at $S=52$ is $0.7041836$
the Delta at $S=53$ can be approximated as $0.7041836+(53-52)0.04429147=0.74847507$
The Delta to be used in the modified Euler method (or Heun Method) is half-way between these i.e. $(0....
Using our good friend Taylor, we know that
where $\Delta_C$ and $\Gamma_C$ are the call's sensitivities and $\Delta_S$ a small change in the price of the underlying asset. In your example, $\Delta_S=1$ and thus,
C(52+1) &\approx 5.057387 ...
The theta for puts and calls at the same strike should be the same, so it seems the SPX theta is somehow wrong.
Edit: thanks @maxim, I see now what the issue is. I think the difference is coming from the fact that the options on the e-mini futures are using the Black formula where the futures price is held constant when calculating the theta. However ...