TL;DR It seems to me that you are trying to find something more fundamental to derive definitions for work and momentum. There is nothing more fundamental than the second Newton's law of motion which directly relates force and momentum (or quantity of motion as Newton calls it)
$$\vec{F} = \frac{d \vec{p}}{dt} = \frac{d}{dt} m \vec{v}$$
Isaac Newton deduced the above equation (law) from a number of observations (experimental results) available to him at the time. It is proved to be true by many scientists after Newton doing even more experiments. If you are interested in discussions like this, then I would definitely recommend getting your hands on "The Feynman Lectures on Physics". The book (in 3 volumes) gives context for many principles that we often take for granted, such as what energy actually means. In short, it is just an abstraction proved to be useful! (See 4-1 What is energy? in Vol. 1)
Here is an interesting article on history of definitions for momentum (vis mortua, dead force) and kinetic energy (vis viva, living force): "D'Alembert and the Vis Viva Controversy" by C. Iltis.
The excerpt regarding kinetic energy:
Boscovich suggested that, if the time coordinate is replaced by the
space traversed and the pressure coordinate by the force which at any
instant produces the velocity proportional to it, a second aspect of the phenomenon is represented. Boscovich, however, explained neither this substitution nor the introduction of the concept of force. The new term 'force' must be interpreted as an entity proportional to the velocity engendered at any instant. If the pressure coordinate is changed to the force and the time coordinate to the space then the new geometrical image producing the velocity would be represented in modern notation as $\int F ds$. We would then interpret vis viva as $\int m v dv = \int F ds$ (where $ds = v dt$). Boscovich does not bring the mass into this analysis.
Why does the same momentum change not mean same energy change?
Simply because momentum and kinetic energy are not defined in the same way: (i) change of momentum is defined as force over time $\Delta p = F \cdot \Delta t$ (impulse-momentum theorem), while (ii) change of kinetic energy is defined as force over displacement (work-energy theorem) $\Delta K = F \cdot \Delta s$.
In your example, for the same acceleration $a$ (i.e. force $F$) it takes equal amount of time to accelerate from $0$ to $v_1$ or from $v_0$ to $v_0+v_1$
$$
\begin{aligned}
v_1 &= 0 + a \Delta t \\
v_1 + v_0 &= v_0 + a \Delta t
\end{aligned}
$$
However, the displacement in the latter case is greater by $v_0 \Delta t$
$$
\begin{aligned}
\Delta s &= \frac{1}{2} a (\Delta t)^2 + 0 \Delta t \\
\Delta s &= \frac{1}{2} a (\Delta t)^2 + v_0 \Delta t
\end{aligned}
$$
For the same force $F$ (i.e. acceleration $a$), $\Delta t$ is the same in two cases which means that change of momentum will also be the same, but the displacement $\Delta s$ is greater by $v_0 \Delta t$ in the second case which means that it takes more work to reach speed $v_1$.
Why so? What is happening to that difference in energy in the two cases... -same forces, same change in momentum, just a small difference in the experiment in terms of when the forces are applied...
In your second question you are trying to integrate the force over time, which is the definition for the impulse $J$ and not for the work $W$. If you still want to get the work by doing integration over time then you have to do the following:
$$W = \int F \cdot ds = \int F \frac{ds}{dt} dt = \int F v \cdot dt$$
In other words, you should actually integrate $F v$ over time instead of $F$. This is why it matters what is the object velocity when you apply force $F_1$ and $F_2$ in your second example.
Best Answer
To throw far, one must throw at high velocity. The mechanics of the human body allow for the fastest velocities through the generation of high torque, rather than linear force. In other words, the high velocities are achieved by pivoting the body around an anchor point to generate rotational acceleration. This allows one to more effectively utilize one's entire body strength, including the elasticity of various ligaments, as is evidenced in the fastest baseball pitches. Clearly, such rotational motion can be best achieved when launching the object from one hand, rather than both simultaneously.