Those are very long, but in general what you are looking for are (at least colloquially) called
machine tooled pin headers
"Machine-Tooled" generally means it's made with something along the lines of a screw-machine. It's a machining process, rather then a stamping process, as is common for rectangular pins. However, it also means they do cost more.
Off the top of my head, I know Mill-Max makes them, but they are fairly expensive (mill-max is high-end/mil-spec. On the other hand, 1 strip of 40 pins may be ~$5, and considering that would make 20 jumpers, cost may not be too big a deal.
I would imagine that other companies make similar parts, but I don't know them off the top of my head.
Digikey carries at least some varieties of them:
The place to look is the "Rectangular - Headers, Specialty Pin" category.
It's also possible the jumpers are manufactured using the pins from some connector, without the actual connector housing. In this case, you're on your own. I don't think there are really any resources for looking up connectors by their pins.
However, the magic words here are again "Machine Tooled". Machine-tooled pins are the round variety (e.g. they are machined, not stamped or moulded).
It is a long question, but better than a short one, as you've shown your own research.
1) Solar cells. If you're stacking your own ones, stack 9 of them and get the 4.5V of the original circuit.
2) Battery charging. Batteries are the only thing you've left out of your spec. This is an area where the circuit design relies on cutting a lot of corners. In theory it might be out of spec, if you were to put 4.5V at 280ma through AA NiMH cells indefinitely. In practice, you don't get full sun all day, you'll be using it indoors, and you're not going to get optimal power transfer from the cells, so this isn't going to cause problems.
3) Diode. It's just a regular diode, not a zener. Current through it is actually determined by the battery and right hand side circuit, not the solar panel - the transistor is off when the panel is generating electricity. The original 1N914 will be fine. 1N4004 will also be fine.
4) Resistors: not a precision component here, use whatever meets your cost constraint. 5.1k for 5k is fine.
5) Wire: not critical. Your ebay link looks suitable. Thinner is better for the toroid.
6) Transistors: stick with the exact part numbers. Design may rely on specific parameters.
7) LED: again, this circuit relies on cheating. Normally a white LED won't run from two NiMH cells. The joule thief part provides a boost converter that gives small pulses of higher voltage. It doesn't have the capacity to provide a lot of current at that voltage. In combination with the pulsing this means there should be no risk of damaging it.
(A proper analysis of this circuit would be good, if nobody else supplies one I'll do it in a few days).
Best Answer
For short distances (e.g. < 0.1 m), stranded or not makes little difference for signals between ICs. I2C is not 100 MHz - max 2 MHz or so. In these cases, the greatest loss is from parasitic capacitance.
Strands touching or not makes no practical difference -- signals between ICs have enough noise margin to make any difference imperceptible.