Oberlo offers some pretty sweet automation options like one-click order processing for dropshipping products: https://apps.shopify.com/oberlo
You can also auto-fulfill orders through Amazon using Order Automator: https://apps.shopify.com/order-automator
I think our app, EZ Fulfill, may be able to help. Our app allows you to auto fulfill your Shopify orders with tracking numbers. You can also use the app to send email notifications to your customers with their tracking numbers.
You can automate the process using FTP, SFTP or Dropbox. We allow you to pull your CSV or Excel file from your FTP/SFTP or Dropbox server as frequently as every 1 hour. The majority of our customers have their fulfillment partners export their orders to CSV after fulfillment and upload them to one of the above services. This allows you to automate your fulfillment process.
I hope this helps and if you have any further questions about the app just reach out and let us know.
I think the shipping time is depended on your shipping option selection. There are many shipping time option on Oberlo that you can choose to decide how long your orders are shipped to customers.
Aftership is one of the best apps that helps you automate tracking and delivery. You can read more about it as well as about how to automate your dropshipping business here.
Hope it helps!