Removal of absolute signs

You don't ignore the sign. Let's look at the original equation, and notice what's happening at $v=10$. The acceleration is zero: $$\frac{dv}{dt}=10-0.1\cdot 10^2=0$$ Since you started with $v=0$, you can see that the object accelerates, initially with $dv/dt=10$, but then the acceleration starts to decrease, as the object approaches $v=10$. After that the velocity can't increase any more. So with your initial condition, you have $v<10$. Then the sign of the expression in the absolute value is always positive. Note that if you would have started with an initial velocity greater than $10$, then the acceleration would have been negative, but always $v>10$ (getting close to $10$ as $t\to\infty$). Then you would use the negative sign when you explicitly write the absolute value.