The Edit Layout button should not be greyed out. Could the file be protected by a password and you have it open with a lower level password?
But adding "several identical drop-downs to the template - same values in all the fields." does not sound like the best database design. You would not do this in Layout mode in any case as you would need to define additional fields using Manage | Database | Fields before then adding the new fields to your layout.
This does not seem like a good approach as the need to have several fields that all record the same type of data suggests that you will likely be much better off adding a portal to a related table where each "field" is now a record in the related table.