Having a single area path for the entire product doesn't make sense. This value will usually depend on the feature, as area paths define the area of code impacted by a work item. This helps with evaluating code churn and test coverage. It needs to be defined per-feature (and per-story).
The iteration path is akin to a sprint, but could also be broken out by other factors. If Aha sets a feature to the default iteration path, and your team in TFS isn't configured to view that iteration, then the team will never see the work item you created and sync'd over. I'd suggest having iteration path defined on the feature or story, but have it default in a value based on the Release in which it was created.
This can be achieved by setting up multiple TFS integrations for your product.
Through settings, you can set up multiple integrations. For each feature, it is possible to select which TFS iteration you would like to send the Aha! feature to.