Work done is defined as the dot product of force and displacement.
However, intuitively, should it not be the product of force and the time for which the body is acted upon by the force (force * time) because while time is independent of force applied, displacement is not.
Were these formulae (for work and energy) actually derived based on some physical understanding or are they just constructs to understand forces better?
Answer
Work done is defined as the dot product of force and displacement. ... should it not be the product of force and the time
Were these formulae (for work and energy) actually derived based on some physical understanding or are they just constructs to understand forces better?
Neither of the two. Most formulas and definitions have an historical motivation. The issue is too complicate to fully discuss it here. There are lots of definitions which are not 'rational' in science, as the first explanations of phenomena where based on misconceptions and some are the cornerstones on which a skyscraper has gradually been erected.
You surely know the names of the orbitals of an atom: s, p, d, f. They are derived from the description by early spectroscopists of certain series of alkali metal spectroscopic lines as s harp, p rincipal, d iffuse, and f undamental. It would be more simple and rational to call them in any other way: a, b, c, d, or 1, 2, 3, 4 or , even simpler, to identify them with $l$, the angular momentum quantum number: 0,1,2,3. It would be rather easy and unpainful to change these definitions (but also physicists are subject to the 'force of inertia': see the comment by John Rennie here), whereas it's extremely difficult to alter the fundamental definitions/ derivations, just like changing the left-hand traffic in England: you ought to change all road signs overnight and, what is worst, scrap all LHD vehicles.
You may find a full and detailed description of how energy was first discovered and neglected in my post here and feel proud that you have had the same ideas of Leibniz.
Leibniz had suggested the more logical, rational and natural integration on time. Now, as you surely know, that is not possible anymore since $F * t$ is defined as momentum.
Another historical reason/ justification related to this is that the concept of energy was understood very late, suffice it to say that, to date, it is not yet considered a fundamental concept and has no own unit: you probably know that the SI has seven base units and energy is not among them. It is really amazing and disconcerting: *the most important concept in the whole universe is a [derived unit]*(http://en.wikipedia.org/wiki/SI_derived_unit) derived in various ways from derived units.
The term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis as "weight lifted through a height", which is based on the use of early steam engines to lift buckets of water out of flooded ore mines. The SI unit of work is the newton-metre or joule (J). (wiki)
Another reason is that the definition was modelled upon the force of gravity, which was for a long, long time the only force that was understood and math- described, that definition, $F*s$, of course, is appropriate to describe it: if you want lift 1 kg by 2m you spend energy that's double the energy to lift 1Kg by 1m. This peculiar definition produced also the 'paradoxical' [see here] (The physical definition of work seems paradoxical) consequence that, in some circumstances, you have spent a lot of 'energy calories' but you have really spent no 'energy work'
But, coming back to the derivation of the formula, after all 'integration on time' is not even necessary as there is not really anything much to be integrated, it is the simple geometric formula you need to find the side of a square: if you consider the unitary mass the real, 'true' relation between entities reveals itself in all its stunning semplicity: $$ v = p = \sqrt{E}$$ We might even drop the 'k' since not only $E_k = W$, but it is also same energy as thermal, and EM etc. energy.
Physicist are probably accustomed to it and do not bother, but it is really puzzling to see the energy of light being related in some way 'equivalent' to the dot product of $F *s$. It would be sensible to replace/integrate this definition distinguishing between work done (energy spent) and work done on something (mechanichal work), and call difference wasted energy/work: $W_d = W_{mech} + W_w$
You'll find more details about wasted energy and wasted energy in a torque at the question linked
I hope this is enough to appease your disconcert.
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