Our apartment is in a complex, does this policy cover damage caused to neighbouring apartments in event of fire, smoke damage etc initiating in our apartment?

There are two possible routes should this happen depending on whether the other party(ies) are insured. If they are insured they would claim through their own insurer, who may revert to us for the losses. If however the other party(ies) are not insured they may sue you and any claims settled as a result of their action would be paid by us.