Should the nursing home provide insulin syringes or should the pharmacy bill Medicare Part D for them?

You'll have to check your policy, but Medicare Part B pays for some equipment. Part D may not, but go to www.medicare.gov and you should find the answers you need. This PDF will tell you what combination of benefits you'll likely need for diabetes products.
http://www.medicare.gov/Publications/Pubs/pdf/10050.pdf

Carol