Since I have the free version of Windows 10 pro, can I still reinstall
it a new?
Just download the Windows 10 .ISO and when prompted for a product key, skip it that step, once installed Windows 10 will automatically activate because of the original digital entitlement you received when you performed the upgrade. Windows 10 for this purpose, considers the same motherboard, the same machine.
Does the windows 10 installation offer disk management and partition
like the older Windows installations used to?
It uses the exact same tool. If it didn't contain the tool it wouldn't be able to create partitions automatically.
How should I reinstall Windows 10 x64 the easiest but still remain on
There isn't a "free" mode. Once you perform the upgrade once, you have a permanent license to Windows 10, which can be installed on the same number of devices, as the previous version you upgraded from. I have previously answered the question about how the upgrade works here.
There is absolutely no difference between the version of Windows 10 you receive if you purchase Windows 10 from a retailer or accept the upgrade when prompted while running an eligible installation of Windows. The only difference is that you won't receive a unique Windows 10 license key. User who upgrade to Windows 10, through an eligible installation of Windows, are activated through a digital entitlement system. Users who purchase a retail license from Microsoft actually receive a Windows 10 license.
Starting with the Threshold 2 update, Windows 10 will both accept an eligible Windows license key at installation and activate with an eligible Windows license key. Windows 10 RTM (Release to Manufacturing) does not have this capability.