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).
EDIT:
It's been pointed out in the comments that I've phrased this rather misleadingly, let me take another stab at this with the following caveat: while I'm confident I have the following set of interactions correct, but I'm uncertain of what exactly the terminology should be.
As a person, you are largely a radio wave sink. In addition to the myriad of electron sources I've pointed out below, you're also continually rectifying radio waves into extremely small AC currents/eddies. In effect: you are a big, really ineffective antenna.
So after the electrons on the surface of your skin which have built up drain off as DC, the difference between yourself and ground will fluctuate in time with the radio waves you rectify. In the US, this is dominated largely by 60HZ, the frequency of the AC current running to all your sockets.
Original answer:
"Super bright" LEDs or other high-efficiency LEDs are designed to turn a very small amount of electricity into light. Since the human body acts like a big capacitor, and you exist in an environment awash in (harmless) radio waves and electrons making a mad leap through the air, or being pulled from your clothes or fuzzy carpets: you tend to build up a bit of charge!
In most environments this bleeds off through your contact with less charged substances, and sometimes simply through the less-charged molecules in the air (especially moisture in the form of humidity).
If you give that charge a different route to earth ground, such as through and LED leg, you'll bleed off that charge a bit more quickly and put a small amount of electricity through the leg. This is enough to get that extremely high-efficiency element to glow very dimly!
Now as for why it work on either leg: there's a threshold of voltage you have to put across a diode to get any electricity through it (silicon is generally about 0.7V at room temperature, gallium arsenide is around 0.3V, etc) and the electricity building up on your skin is AC. What end up happening is that the LED is getting a tiny AC current around this threshold, and the diode part of the Light Emitting Diode means that the current only flows the "right" way. Thus, light!
Hope that helps.
Best Answer
No. Definitely not.
All LEDs degrade with time and having life times of under 10,000 hours is common.
LED YSM-1288CR3G2C data sheet here
This is almost certainly not an OLED.
Your Red LED has a Vf (forward voltage ) of 1.9 - 2.4 V
The Green LED has Vf of 3.2 - 3.5 V.
The Green LED is probably industry standard InGaN.
The Red LED is probably industry standard InGaAlP
As a guide to lifetimes, very very very few 5mm White LEDs have lifetimes to 70% of initial output of more than 10,000 hours.
This is despite the widespread claims for 100,000 hour lifetimes.
Any LED which achieves lifetimes o > 50,000 hours either comes from one of the top ~= 6 LED makers or uses their patents.
They say Red: 4.7% - 8.3% / 1000 hours Green: 11.4 - 15.3% / 1000 hours
The manufacturer is unusually brave and honest in quoting such a poor 1000 hour lifetime. This matches reality for many Asian sourced LEDs but most related data sheets do not tell the truth.
However - this loss rate is acceptable in many applications where the LEds are not illuminated "all day every day".
A good quality LED will have minimal loss of output in 1000 hours and many have MORE output at 1000 hours than when new.
The industry standard data sheet test period for output is 1000 hours with less than ?10%? decrease from initial output level and this is essentially worthless as all quality LEDs meet this with ease.