What's the rationale behind these warnings about extension cords?
Let's say you're extension cables are 1 mm² cross-sectional area. 


The loop resistance is 33.6 mΩ/m.
A 25 m cable will have a loop resistance of 840 mΩ.
The voltage drop at 11 A will be given by V = IR = 11 x 0.84 = 9.24 V. On a 220 V supply this is a reduction of 4.2%.
Power dissipated as heat in the cable is given by P = VI or P = I²R = 11² x 0.84 = 102 W. This is a waste of 4.2% power (as we would expect from the previous calculation).
String four cords together for a 100 m run and you'll only get 220 - (9.24 x 4) = 183 V.




The situation is a little more complicated than the calculations above suggest because when you add in the cable resistance the current will reduce too. For a 2.4 kW, 220 V load we can calculate \$ R = \frac {V^2}{P} = \frac {220^2}{2400} = 20.2\ \Omega \$. Add in our 100 m loop cable resistance of 4 x 0.84 Ω and the total resistance is now 23.6 Ω resulting in a max current of only 9.3 A which at 183 V (calculated above) reduces the power to VI = 183 x 9.3 = 1700 W which is only 71% of the rated power. The "square" term in the power equation makes the power loss proportional to the square of the voltage loss.



Now to your questions:


  Don't use an extension cord longer than 25m. ... and the tool wouldn't be able to operate at full power. But this argument doesn't convince me. 


Be convinced by the maths.


  All the extension cords I've ever come across have at least 16A limitation, so there is quite a reserve still. 


See if you can find the cross-sectional area for those cables and recalculate.


  In addition, the power has to travel much, much further to get to my house, and even after the meter it still takes probably close to those same 25m to get to the power outlet.


Ideally your local transformer will be fairly close to your house and, in Ireland, the connection will be using at least 25 mm² cable. You can get some idea of the source "resistance" to your house by monitoring the voltage while you switch on and off a large load such as an electric oven and all the hobs.


  Don't daisy-chain extension cords. ... the rest seems to me to be practically negligible. 


It may be tolerable (depending on your load) but it's not negligible.


  Is there really a significant contact resistance? 


This is usually not a factor.


  Make sure that the power cord is entirely unrolled. 


Our calculations showed that the power dissipated in the cable is 100 W. With the cable tightly rolled the coil temperature will rise. Imagine you wound the cable around a 100 W filament light bulb: what temperature do you think would be reached? Would it get close to melting the insulation?



Figure 1. Source: Why is it dangerous use a coiled extension cord.


  My suspicion is that it has to do with inductive resistance. 


The term you are looking for is "impedance" for AC resistance. Actually there will be almost zero inductance as the live and neutral currents are in opposite directions and cancel out. I used this trick in work when I needed to create a test load for a 30 A, 30 V 50 Hz supply. We first wound a coil of 1.5 mm² cable of the correct resistance but it was really hopping when we powered it up. When we unwound it, folded in half and rewound it there was no buzz.



Figure 2. Inductance cancellation. See my answer to the question linked above.


  But is it really that strong? ... You need a lot of neat, tightly-wound loops of wire. 


Or a high current in a low number of turns.