You might test the fixing of NT fibrest in server properties as per the KBA http://support.microsoft.com/?id=303287 link. Also look at the SQLAgent log in this case for more information,as it writes there.

I don't think doing so, you would restart the log shipping process again, only thing to make sure is their will be no LSN mismatch. You can refer what was the last restored Log Backup and then restore immediate Log Backup to your secondary server and then restart your log shipping.

If the log shipping process say backup of log is under progress then you will lose all uncommitted transactions and that are failed to written to log. For the next time when log shipping job starts to secondary server it will fail with log mismatch, you might need to apply full database backup in order...