Well, this is something squarely in my wheelhouse! Maybe I can help.
You should first check your short-term disability policy (ask HR for a copy), because some of them have a 5- or 7-day elimination period, which is the number of days you have to be out before the benefit starts. So if the first week is not payable, it makes no sense to file a claim for just one week.
The amount of time off you need depends on the type of work you do. If you have a physical job (lots of standing, walking, lifting/carrying greater than 10 pounds), then your doctor will probably recommend you take more time off and the claim will be approved. If your job is mostly sitting at a desk, your doctor will have to give specific reasons for your inability to work. Depending on your procedure, most doctors already have a set number of weeks they will sign off on for recovery.
Either way, you aren't tied to the initial number of weeks you requested: you can always return to work sooner if you feel well enough, or you can apply for more time off if there were complications.