I have copied a database running in archive log mode onto another server. When starting up the database and issuing the command alter system switch logfile I find that the owner of the logs is root. Will this cause me a problem later when users are logging into the database.
Don't know - what are the permissions on the log files? A UNIX-based Oracle-related account should be the owner of ALL your Oracle files. "root" is a privileged account in UNIX, just as sys/system/dba-enabled accounts are in Oracle. You need to fix the ownership of your Oracle files. If you don't know how, look up the following commands:
Again, based on what you provided, the answer is "don't know" - what if the permissions are 644 for root? Oracle can read, but not write. Did alter system switch logfile work or not? If the Oracle owner can write to the files, then ownership isn't necessarily a show stopper.