Yes, the batteries have an internal BMS; so it won't harm them.
Next time just try to get a higher power version (more Milli-amps), 6V, without diode.
Just as long as you connect the leads correctly.
I don't know about solar panel discharge; I should find a way to test it though.
DX sells Gameboy DS batteries for very cheap (like $3). You could try to connect one with panel, the other without to see if it loses energy over time.
I gave up on the solar panel, as I waited for 9 months, before cancelling the order (I had already sold the Jetbook, so no need for solar panels for a while).
ALthough I might want to install one on my EZ-Reader Pro.
15 minutes of reading time on the Jetbook equals about 15 hours on the EZ Reader PRO (has e-ink), and by the time you reach those 15 hours, the solar panel most likely has charged it even more. It could technically provide the EZ REader Pro with an infinite amount of energy, just as long as there is light.