Rooting and carrier unlocking are completely different things. The confusion arises because there is bootloader unlocking (commonly a prelude to rooting) and SIM unlocking (allowing use with any carrier), but they are entirely unrelated. Rooting will not by itself network unlock a phone, and network unlocking does not give you root.
The phone will still work as a phone after rooting. But some apps check for root and will not run on a modified phone (banking apps, payment apps, some media apps). Some root methods and tools allow you to hide it from some of these apps, but if any of these things is important to you you should check that you have a method that's known to work with the particular app you care about before proceeding.