The inverter has a male "car cigarette" type adaptor and the battery just has a standard output designed for crocodile clips.
We recommend you get a female version to connect to the battery then on to the inverter.
We can not recommend that you cut the adaptor off and connect to the battery but this should work too.
Yes - this will double the duration the system will run for between charges. Alternatively you could double the panel size which would charge the existing battery twice as quick and may prevent having to do any mains recharging of the battery.