This does not happen automatically. The time is the original publish time. To change this when editing a pot look down the right series of blue headings and find ‘Post Timestamp’. Check the box, change the time.
This behaviour is normal to the other version of WordPress – it is not just a .com feature.
Perhaps showing the time is theme dependant I do not know. It is not a feature on the blog templates I am using but there’s always the manual method available to you. Simply type the time of completion at the end of each of your posts and your readers will be able to see it.
Showing the time is theme dependant. You can’t change it with the setup we have here since we can’t edit the files. (If you were able to, then you would be editing them for everyone who used that theme and the same if someone else edited it on you.)