The $listDays option is used to limit how many days the List module will use to generate the list. If you use it in conjunction with $noOld, you can then set it to always display the next 7 days.
There's no built-in option to be able to generate the list using a single date header. I was able (and others too) to use the Blog mode in a loop to generate that kind of look. I've polished my solution to limit the amount of code it generates and can also bypass empty days. If you are interested in buying it, contact me using the link in my signature below.
As for displaying only a week using the TOC, that's not possible.
EPC Tutorials... We're here to help!
Offering custom integration services. Contact me here
* Not affiliated with EasyPHPCalendar or NashTech Inc.