1. What is the curly-looking thing labeled "22uH"?
That is an inductor. 22 microhenries. Good electronic stores might carry them.
2. There are no labels on the transistors in either circuit. I can tell that one is PNP and one is NPN but I can't tell which, nor what type is needed.
Yes, there are. 2N3906 is the PNP and FJN965 is the NPN. The first is quite readily available, but I have never heard of the FJN965.
3. What's a "1 M" resistor?
1 megohm. One million ohms.
4. The original schematic shows the solar panel and D1 and D2 diodes. Yours only shows one diode and no solar panel.
EDIT: Oh, wait, you had only redrawn the oscillator part. I'm still rather confused though, because I'm not sure how yours fits into the big picture. If it's not too much trouble, could you draw the full circuit with labels? That would be great.
Also, this oscillator only gives 1.2 volts to the LED. If I needed more, could I add batteries in series with the one shown without any further alteration of the circuit?
No, it gives more than 3.5 volts because it lights up a white LED. It is in the form of pulses.
Here is the circuit again with the solar panel in it.
Is there anyone who can help you with this? It is not a simple project, especially if you want to modify it.
You will need test instruments to be sure it is working OK.