Can someone let me know linearity, time-invariance, causality, memory characteristic of the system?
There's an error in your 3rd section (linearity) - you replace $y(t)$ with $y_1(t)$ on the left side of the equation but not on the right side. You should have
$$y_1(t)=x^2(t-t_0)-\frac{1}{2}\frac{dy_1(t)}{dt}$$.
Now you see that $y_1(t)=y(t-t_0)$.

As for memoryless - start with your initial equation and integrate both sides. Now you have
$$
\int y(t)dt=\int x^2(t)dt-\frac{1}{2} y(t)dt
$$

and rearranging gives
$$
y(t) = 2 \int x^2(t)dt-2 \int y(t)dt
$$
Does that help?