Can I select the default status that a record must be set to when I reopen it?

Yes, you can do this by selecting a Re-open status.

Re-open status is the value that a pre-selected state picklist field with closed states configured, takes when a record is re-opened. The re-open status must be configured for all the modules.
For example, the pre-selected state picklist field in Contacts is Status. When you select ‘Cold’ as the re-open status in Contacts, every time you re-open a contact, the Status field's value is set to ‘Cold’. 
 

Follow these steps to select the default re-open status:

1. Log in to your CRM account.
2. Click the User Menu on the top right corner of the CRM screen.
3. Click Settings.
   • This will take you to the Settings page.
4. Look for the Module Management section.
5. Select Module Layouts & Fields.
6. Choose Contacts from the Select Module dropdown.
7. Choose a value from the Select re-open Status dropdown.
8. Click Save.

To learn more about closed states, click here.