Q: iTunes 12.5.1 and iOS 10.0.1 update

Yes, unless you were a lucky one who installed iOS10 over the air and found it did not require the newest iTunes for support.  A regular iOS upgrade will usually require the current version of iTunes at the time of the upgrade release.

iOS 10 requires iTunes 12.5.1 or newer which in turn requires OSX 10.9.5 or newer.  Software Update only checks for updates for the current system version you are running but that itself may be out of date.  It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system you will not be able to sync this phone to your present computer. OSX 10.9 and 10.10 are no longer available. Check this article on upgrade options: If your Mac is not Compatible with Sierra but you Have Not Upgraded to El Capitan  - https://discussions.apple.com/docs/DOC-10692  El Capitan (OSX 10.11) and Sierra (OSX 10.12) may run slower on older machines and require additional RAM purchase (4 GB is a realistic minimum and many recommend 8GB).  Doing a big jump in system versions is also more likely to affect old software.

Snow leopard to el capitan, will it cause my macbook to be slow?  - https://discussions.apple.com/thread/7412959

Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/

Hi,

OS X 10.9.5 or newer and iTunes 12.5.1 is required for syncing with iOS 10.

Jim