Should Development Teams work the complete 10 days of a Sprint?

I'd suggest discussing that with your development team. Does working all the way up until the Sprint Review to meet the Goal compromise the quality of the increment? 

Also, what impact might working on new items all the way up until the very end of the timebox have on sustainable pace? 

What is the goal in Scrum?

• Delivery of value to the business/customer?
• Meeting the Sprint Goal?
• Completing the Sprint Forecast?
• Optimization of each Development Team member for the length of the sprint?

Is there a problem/issue here?   What might the result look like when a Development Team works on Sprint Backlog items up until the last day of the sprint?   Has anyone asked the Development Team how they feel about it?   

In a ten day Sprint ,should Development Teams work on the Sprint Backlog items from Day 1 to Day 9 and leave Day 10 for the Review and Retrospective or should they work on Sprint Backlog items from Day 1 to Day 10?

From your reading of the Scrum Guide, how long might the Review and Retrospective be expected to take on Day 10?

In your example above, does planning and grooming count as working on the sprint items? Does collaboration with other devs or technical discussion along the way count as working on the sprint items? 

We need to have a clear understanding of what you meant by "working on the sprint items", is this actual sitting on the computer for 9 days or collaboration or meetings are part of it. You need to also communicate this to your team so all are aligned.