Currently in the absence of a date parameter in the query string it uses the current date as the start date and goes 7 days into the future from there. If the next or previous link is clicked then a date parameter is in the query string and it uses that.
It would be fairly easy to add a setting for an offset value by a number of days, in other words there could be a setting with the value 7 which would mean in the absence of a parameter add 7 days to the current date and use that as the start so by default it shows the next week instead of the current week unless the user navigates.
Making it go by a day of the week would be much more difficult to solve. How do we decide if its the previous or the next monday from the current date for example?
It is currently possible to customizing the date format of the header on weekview/dayview. See the article Event Calendar Pro Tips and Tricks. If you want the day name included for example you could use: