Can I hide the dropdown in the popup form?

Yes, the easiest way to do this is to add some CSS to your signup form settings:

1. In your Back in Stock account choose Campaigns, Signup form from the top menu
2. Choose the Color and Styles tab.
3. Add the following to the Custom Styles field:
     
select#variants { display: none; } 

4. Hit Save form settings.

That will hide the dropdown in the popup form.