SQL Server Backups - A couple of questions
(1) Yes, I usually have the backup process by itself. I would not be doing much of anything during my backup time if I can. You might have it take the backup and then do the update on stats. As it sounds it seems you are running two jobs (1 for backup, 1 for update stats) at the same time?

(2) Do you copy the backup to tape or other disk storage? If so then I usually clean up the files before I create new backups locally. If not, then if I am scraping for storage space I would consider zipping the backup file prior to the new one being created. (That is if you cannot enable compression on the backups, as @Simon suggested, this would save some space as well.)