The frequency and all that will depend on your backup and recovery needs.

So long as your full backups are going onto tape or something of that sort, overwriting the old one with a new one shouldn't be a problem.

The only reason to keep old diff backups is if you want to do a point-in-time to something before your most recent diff backup. In your case if it's Friday, and you want to do a point-in-time to Thursday afternoon, it'll be easier to restore the last full, restore the Thursday morning diff, then use Thursday log backups to bring it to the desired point, than to restore the last full and then restore almost a full week of log backups. If that's not going to come up, then there's no real point in keeping the old diffs once the new one is written.

Log backups, definitely don't overwrite those. That would defeat the whole purpose of doing them in the first place. Just make sure you dispose of ones that are aged out.