How do low power draw devices not cause problems?
The last sentence in your first paragraph is wrong.  A device drawing load on the wires does not reduce their load, it does not prevent them from heating up.  Actually, putting a 2 amp load on a pair of wires increases its load (by 2 amps) and causes it to heat up (by W=I2R or W=2*2*R watts, with R being its resistance in ohms). So a 0.1 ohm wire is now making 0.4 watts of heat when it wasn't before. 



If it helps, recognize that some loads have electrical characteristics which naturally limit how much current they flow at a design voltage.  We supply them that designed constant-voltage, and they sip only the current they need.  For instance a resistive "1500W" heater will limit its draw to about 12 amps.  

Not all loads have this ability.  Some loads cannot limit their current and will flow uncontrollable amounts of current if supplied a constant voltage.  A great example is a fluorescent tube, which is why fluorescents have ballasts. 

However if some outside force limits how much current they can draw, then they behave in reasonable ways: the voltage that they "drop" (consume) at that current is reasonable, and produces useful work -- like light from the fluorescent tube or LED emitter. It's possible to build a constant-current power supply which will forcibly limit current to spec. This power supply need not be constant-current; it can be varying-current, the point is, it must limit current because the device it's attached to can't. 



How do you hook up multiples of these?  

For constant-voltage devices of the same working voltage, for instance a bunch of iPhone chargers that want 120VAC -- you supply them all spec voltage, and you hook them up in parallel.  Each device "sips" the current that it needs, and no more.  The power supply must push enough current to work all the devices. 

For constant-current devices of the same working current, for instance several bare LED emitters that want 350 milliamps -- you supply them all spec current, and you hook them up in series.  Each device gets its design current (because everything in a series loop gets the same current), and each one "drops the voltage" it needs to function.  The power supply must push out enough voltage to work all the devices. 

What happens if you take a constant-voltage device like a small heater, designed to run at 117V and which will tend to draw 350ma... and you put it in series with constant-current load (like an LED emitter with rectifier) that is designed to draw 350ma and drops typically 3 volts? That will work, actually.  The constant-voltage heater will act as a current-limiter for the LED, keeping the LED in spec at 350ma.  



In mains power, you are dealing with devices designed to be constant-voltage, so you connect them in parallel.  

Wire-nuts will flow unlimited amounts of current uncontrollably, so they simply do not work as constant-voltage devices.  They are better thought of as constant-current devices with very small voltage drop - like the LED in the LED-heater combo discussed above.  They must be in series with a current limiting device of some kind, like a constant-voltage appliance which is able to limit its current to within the spec of the wires.