If I add another line to my account, will this change my contract date?
It won't affect the contact date on any lines you currently have, but if you purchase a subsidized phone the new life will be under a new 2 yr contact.
What if I use an old phone that I am no longer using for the new line?
If a VZW branded phone, then you could use it on the new line and that line would be a month to month line
Accounts do not have contract dates. Individual lines do. And they only have contracts if one is purchasing a phone with a subsidy.