Hello Joao
From your name & grammar I assume English is not your native language
1) battery should be 12.5 V ( or more ) when fully charged .
When you try to start the engine it should not drop any lower than 10v while the engine is cranking
When taken off the battery charger the battery voltage should read 13 V or higher then over the next couple of hours it should drop down to around 12.5 V and stay there for weeks if not connected to the mower .
If the battery fails either of the above it is bad .
2 ) not sure what you mean
3) from that i think you mean that it will start by pulling the rope but not start when using the starter motor