Our roles are not purely product development. We also provide '1st-line support' for internal & external customers, and any of these tasks, by their very nature, will always override any product development priorities.
How can we use SCRUM's sprints to help us manage product-development and support issues?
You might want to take a look at kanban or scrum-ban. I'm not a fan but it may work better for your scenario where distractions and interruptions may be unavoidable. Ditch the sprint but still keep a prioritized backlog. Rather than tracking and measuring spring velocity, measure latency in every phase.
http://leansoftwareengineering.com/ksse/scrum-ban/
I would recommend taking a step back though. If you want to be an effective agile team you need management buy off... why is the development team doing first level support? Do you have a strong scrummaster that is able to insulate the team from distracting internal customers? I don't know what your support volume is but I'd play with rotating through your team members into an impediment magnent position where they take all support/internal customer flack for a week at a time, allowing the other members to focus. In any case, pick a scrummaster... rotate team members through that position until you find the right person for the job.