The first thing you really have to determine is whether the circuit they are going to be plugged into, in the house, has the capacity to TAKE the extra draw. Heated waterers/bases and heatlamps draw pretty significant current -- like between 120-250 watts *each* -- and especially if this is being multiplied by 3 coops, that could be a problem depending on what is already on the household circuit.
Running extension cords (esp. multiple ones) across the yard is really not a very safe arrangement, neither from the perspective of humans or animals getting shocked or from the perspective of an overloaded wire or partly-pulled-apart plug connection starting a fire. Coops *do* burn down from things like that; people *do* get zapped hard and occasionally seriously injured or killed.
By far the best thing would be to either a) trench in a PROPER electric line yourself and then hire an electrician to do the final connections for you. Or b) go without electricity, haul water as needed to keep it thawed (using large waterers with insulation will lengthen the time it takes to freeze), and use battery or solar-operated shed lights as "work lights", and solar yard lights if you really want to extend daylight for laying (although I question whether the latter is worth the aggravation).
If a person is *going* to use extension cords, at least use the heaviest-duty ones you can find, the 14-gauge 'heavy duty' ones, and use a single long cord rather than multiple shorter ones strung together. Anything joined should be knotted (but not so tight it pulls the plug connection ajar) and ideally wrapped with electrical tape or something like that, and set up OFF the ground well above any level of flooding or snow, with the junction the highest point so that any moisture getting on it runs off to either side, and covered with a bucket or whatever. It is still not a great idea, mind you.
The octopus of two cords in series branching out to three different coops from a multi-outlet adapter particularly worries me...
Good luck, have fun, be *safe*,
Pat