Deriving $F = ma$ - Newton's Second Law of Motion

1. Yes. It should be: $$frac{dp}{dt}=frac{d(mv)}{dt}$$ I'm using $d$ instead of $Delta$ because I am thinking about the limit where the changes in $p$ and $t$ are very small. Then these are called infinitessimal changes, and denoted by a $d$.

2. Usually, when one considers simple problems in Newtonian mechanics, what one does is study a given object with a fixed and constant mass, $m$. This means that $dm=0$ at all times by definition. We have, using the product rule (which only holds for infinitessimal changes): $$frac{d(mv)}{dt}=vfrac{dm}{dt}+mfrac{dv}{dt}$$ Can you guess what happens to the first term on the right hand side of the equal sign?

You are right, there is a $Delta$ missing in front of the $t$.

$Delta v = v_2 - v_1$. If the mass is not changing, then $Delta (mv) = mv_2 - mv_1 = m(v_2 - v_1) = mDelta v$. Hope that helps.

The equation that includes $frac{Delta m}{Delta t}$ is not Newton's second law. The second law is valid only for systems of constant mass. An equation like that one does appear in the complete analysis of systems of variable mass (like a rocket with its propellent being exhausted from the rear). It is also sometimes called Newton' second law in incorrect analyses of the rocket problem.

Well that's a result using differentiation and derivation.
Have you studied calculus? If not, there is a simple way to look at it.

$$frac{Delta p}{Delta t} = frac{Delta (mv)}{Delta t}$$ (Yes there should be a $Delta t$ in the denominator, too.)

Now, what does $Delta(mv)$ mean? It represents the change in the quantity $mv$. For current situations, your text book assumes that the mass of the body will be constant with respect to time, so it will not change! Therefore $mv$ is like saying $text{(some constant)}v$. Since this constant won't change with time, we can take it out of the $Delta$ and write $Delta(mv)$ as $mDelta v$.

You can't derive Newton's Second law from within Newtonian physics, even its original, general form of F = d(p)/dt. F = m*a is just a special case, not a "derivation". You can derive it from the action principle, but then you would and should ask whether you can derive the action principle. The answer is you can't. From the action principle you can derive the laws of physics (including general relativity, quantum mechanics, classical and quantum field theories ...) at least separately ;-) ), and these laws are found to hold true.