Where is PMS running? On the NAS? If so, then yes, you can log out and turn off the computer. Once the scan starts, PMS will do the rest, so as long as your server is running, the instance of Web that started the update does not need to stick around.
In have not generally found issue playing media while a library scan is running. I have seen in the forums that people post about some performance issues. It’s really about the NAS server and disk performance.
I also generally run the large scans overnight when the activity on my media server is light ( including my shared remote users)
It depends if this is the initial scan creating the library. That can take up some resources. If it is a subsequent scan to add more items, I don’t see any problems.
If you see a problem with this, there is a PMS option that could help.
Settings \ Server \ Library \ Run scanner tasks at a lower priority