EDIT 2013-09-02 - Important - see comments after this answer for some important caveats.
You may want to check out Crashplan. It's a very configurable and free backup solution. You can backup to another device on the same computer, another computer on your LAN, or another computer over the internet. If you wish to pay, you can use their cloud servers, but if not, you can use it for free with your own devices and/or server(s).
I have been using CP for about two years and I DO pay to backup to their cloud servers (although as I noted above, that is optional - I mention it in the interest of full disclosure). It is the most configurable backup solution I have found and can easily handle all the scenarios you mention, and others. OTOH, its UI has some issues. Bottom line, though, is that it gets the job done and the essential stuff is straightforward to understand & operate. I've also found their tech support responsive when needed.
Among the features you can tailor is how long the backups are retained (e.g., your 7-day example), how often backups happen, how many versions are retained, etc. Here's a screenshot to give you an idea of the flexibility.
And here's the Frequency and Versioning dialog:
Note the sliders let you decide how aggressive to be in your backing up (I've chosen the most aggressive settings). For example, I never delete files from the server, even when deleted from the client.
OTOH, I could slide it left and have client-deleted files deleted from the server every week:
As I mentioned, I've been using them for a couple of years and despite some marginal issues am reasonably happy. I have no other connection to the company.