Yes, you can get the ids of the records. In the constructor for your standardset extension, you can get the records using the following code:
public List<sObject> myRecords {get; private set;}
public MyClass(ApexPages.StandardSetController ctrl){
myRecords = ctrl.getRecords();
// - OR -
myRecords = ctrl.getSelectedRecords();
//To get the ids of those records, you could do something similar to:
Set<Id> myRecordIds = (new Map<Id,sObject>(ctrl.getRecords())).keySet();
}
From there you can either make updates to myRecords, or use their Ids to generate children records based on those parents. Note, the getSelectedRecords is relevant if for instance you have a related list with checkboxes and a custom button (meaning the getSelectedRecords method will give you the records in that related list that are checked).
Hope that helps.