Your panel takes 9v alkaline batteries that are non rechargable , so I would assume that the panel does not have the necessary charging circuit to support a 12v rechargable.
Only another Optima2Plus RKP will do , and its now obsolete.You would need to replace the main panel and remote keypad together , maybe a Accenta G4 , or Texecom R8 would do.
Check the log to see which zones are triggering and why.Then check your programming , you won't be the first to have problems with the Accenta programming.
The AC output from the transformer should be around 18v.You should then have 13vDC at the battery terminals , if not then your panel is faulty.Check the fuses (with a meter) .It could be that the voltage regulator is shot , but I would just swap the panel for a Veritas or G4.
Your zone inputs are basically a send and receive circuit , so if it expects say 12v on its return terminal and your receiver has 12v on its C or NC then the panel will see no change...its improbable , but worth a check.
Put a small loop of wire in zone 5 , set the system then break the loop , if the alarm triggers , then you know that the panel works and the problem is with the wireless receiver.
If that is the case , check there are no voltages on the C and NC.
Yes , open the box (internal alarm will sound , enter your code to silence ) follow the instructions re:defaulting codes , user and engineer codes will be defaulted.