Regardless of how you run power to the shed, you should run minimum of two supplies, and have your enlarger on its own circuit, and run whatever else on the other. BTW, maybe I should fire my electrician and throw out my wiring manual.
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A good, heavy-duty extension cord can be found among those intended for portable generators. 10-gauge general-purpose cords usually have 15 ampere plugs (the parallel-blade, U-ground NEMA 5-15P). The heavier wire reduces the voltage drop, which is important if you are using both a light source (enlarger, or contact-printing light) and a variable load (space heater with thermostat) on the same circuit. Cords meant for generators, and some intended for construction sites, have 20-amp U-ground (blades at right angles) or twist-lock connectors, and there are even 30-amp twist-lock cords. Of course, you need the appropriate outlet to plug into, protected according to its wiring in the wall, and you might need an adapter at the darkroom end to get two or three 15-amp outlets.
The good thing about this approach is that you don't need to worry about doing your own wiring, and the bad thing is that 50-foot cords like this start between $50 and $100! (Also, you will have no better ground fault protection in the darkroom than you had at the house where you plugged it in
You can calculate the amp requirements of the safelights based on the bulb wattage. 3 safelights, each with a 15-25 watt bulb, should total less than 1 amp. So while the LED lights are a good idea, that's not likely to impact total electrical draw all that much.
Your heater is the big issue. It will draw significant power; more importantly it will draw at a variable rate depending on whether the element is on. Best approach (I think) is to use the oil filled radiator and turn it off while printing. The heater should retain enough residual heat for a while.
Excluding the heater, I doubt you'll draw more than 6 amps for enlarger, safelights, and timer.
You need to be concerned with voltage drop, which will be a function of the size of the extension cord, the wire gauge, the amp draw, and voltage. You also have to consider voltage drop for the circuit you're plugging into. There are calculators online, but you should be OK with a 12ga cord at 50 feet. Voltage drop may not prevent you from getting the print you want, but could wreck havoc on repeatability of results. Of course, you need to be concerned with any other loads on the same circuit; especially loads that might be variable.
The circuit you're plugging into must be GFCI protected; or consider investing in more insurance.
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Regarding safelight(s), safe is safe. You gotta test it yourself with the light sensitive materials you use. I have one double output light at one end of my lab, but painting the walls white (instead of darkroom black) reflects enough light from it into the far end sufficiently for my purposes.