Does a solar panel produce full power all the time?
Short answer : No
First, let's take a look at a solar panel's characteristics, which are always some variation of this graph :
Image source : http://www.pveducation.org
You can see current and voltage and current are linked through the red curve. In practice, it means that if you do not pull current, your solar panel will quietly sit at Voc (open-circuit voltage) and pulling current from the panels lowers the voltage, but increases the total power output, until you reach the maximum power point (maximum of the blue curve), from where pulling more current will reduce the voltage so much that te total power available decreases.
Which means, as @mkeith pointed out, that you can adapt the solar panel's energy production at will.
Now assuming your power conversion works correctly, it should perform both these tasks :
- If the battery is fully charged, provide exactly the power needed for your appliances
- If the battery is not fully charged, provide the appliances power + charging power for the battery, while taking its maximum charge current into account.
What happens to the sun's energy when not turned into electricity?
Short answer : It is turned into heat
As shown above, a solar cell does not produce its full power all the time, but still receives energy from the sun. Conservation of energy tells us that it should go somewhere. Basic logic tells us that there are only two ways : Light and heat. This page from pveducation tells us that it is simply turned into heat, which will then be dissipated via convection, conduction or radiation.