Is it possible to change the status a new order gets?

Yes, by default orders will go into the finished status, but that can be changed by adding a new system setting. This setting is not created on install, so you'll need to create it manually.

The setting to create is simplecart.finished_order_status. For other statuses you can use the same structure, for example simplecart.new_order_status to change the status of a freshly created order before it is marked as paid/finished.

The value of the setting needs to be the name of an existing status. Statuses can be viewed and created under Extras > SimpleCart > Order statuses.