Is a phone with a model number starting with 5 new or refurbished?

You can’t reliably establish whether the phone is new from that first digit alone. A 5 prefix has been reported in connection with third party refurbished iPhones, so it deserves investigation, especially if the seller advertised it as new.

Look in iPhone > Settings > General > About for Parts > Service History. On supported iPhones and software versions, this can identify replaced components, including parts marked Genuine, Used, or Unknown. Its scope varies by model, so it is not a complete certificate that a phone has never been used or repaired. 

Follow up with the merchant and contact Apple Support in your country or region for verification.

If you’re in the US, please call  1-800-MY-APPLE, or outside the US, consult the support articles below. In Canada, call 1-800-263-3394.

Contact Apple for support and service - Apple Support

Genius Bar Reservation and Apple Support Options - Apple

Only Apple or the merchant can tell with 100% certainty. However, every indication is it’s refurbished, has no warranty from Apple, but may have a limited warranty from the refurbisher and/or merchant.

That’s further indication of a refurbished device. What about the parts?

There is no service history with replaced components in the settings

In the "About this device" settings, the warranty information indicates that the warranty period expired in January 2007.

Thank you for your response and for the time you’ve taken! 
I intend to file a complaint with the seller, as they sold me a refurbished iPhone—presenting it as new—without disclosing this fact.

You’re welcome and I hope everything works out for you and that you have a great weekend!