Imagine a bunch of processes using a
/home/user/logs directory. They have some logs open for a longer time.
Yes, under certain conditions. The inodes of the files won't change if you move their parent directory, so the open file handles will still be valid. The conditions are:
Processes usually do not keep files open in logs for long.
Yes, you can move / delete any directory or file which is in use by any process at any time. Heck, you can delete the process' executable while it is running and it will continue to run.
However, prepare for some possible scenarios:
Files will be physically cleared on delete when no process will use their descriptors. You will see the files moved / deleted immediately.