Can I use the battery for supplying Arduino and the rest of the circuit, when at the same time I charge it?
Yes. Your phone works while you are charging it, doesn't it?
The solar panels have output 12 Volt, 600 mA.
Rubbish. That MIGHT be the maximum output on a very sunny day.
Shouldn't I make the Arduino+circuit use less than 600 mA in order to let the battery charging?
You can't make it use less current. Or more current. The only way to control how much power it uses is to turn off things that it powers.
If the Arduino draws more current that the charger provides, the battery needs to supply the rest.