Yes, the idea of the 7 days limit on laylife after ordination was introduced in the Visudhimagga so far as I know. It has no Sutta counterpart. The gist of the idea is that without a means to gain food you would not survive - no supernatural reasons are given for the spontaneous death.
This sutta, as many many other suttas, has a relevance with the dependent origination teaching that many teachers have tried to explain as happening in the space of an instant. In this interpretation jati means to be 'reborn' into the present moment, as one who is happy, angry, sad etc... Yet, when we see jati described in such suttas as above, it is patently clear that what is meant is literally rebirth in the heaven\hell (etc.) realms.
Also, it has been dawning on me over recent years that practically every time the Suttas talk about karma, they refer to modes (and conditions) of rebirth. This is not the same emphasis that karma is usually given in 'pop' Buddhism eg. your car is stolen due to your karma. Granted, conditions of rebirth last through life - for instance one is a rich person if previously generous, one is a leader if one was previously respectful etc.. Yet, Karma seems to really have very little to do with day to day actions.
Therefore, though Karma refers to action, and vipaka the results, I do not see that we have to generalise to mean ALL action - it practically always refers as in the quoted sutta to rebirth.