[SalesForce] Setting default value for datetime field

If you have any record types associated with the object, you'll need to select the default picklist value for those record types as well. That should solve your issue. If it doesn't, make sure your controller class is set to "with sharing" as that also may have an impact on the default value display.

You can use Developer Console's Execute Anonymous feature to accomplish that.
You will find it on Debug drop down list on Developer Console.

Query null values from desired object and update them.

Code Example for Account object:

List<Account> AccList = [select id,customDateField__c from Account where customDateField__c = null];
for(Account a:AccList){
   a.customDateField__c = system.today();
}

update AccList;