Do kittens continue to improve at jobs that aren’t doing any work

The main section, above the census, is where you increase or decrease the number of kittens on a given job. You don't get fine-grained control over which kittens are doing which job, but you don't really need it; removing a kitten from a job using the [-] button always removes the kitten who is least skilled at it (of those working at that job right now), and adding a free kitten always adds the free kitten with the highest skill in it. As of 0.9.9.2, you can remove a specific kitten from their job when you unlock the Census, but assigning still always assigns the most-skilled kitten.

Note that your kittens can be skilled in more jobs than display, so you may not always be able to tell at a glance, but under most circumstances, your kittens shouldn't be changing jobs too often anyway.

Many browser games do not run while you're in another tab. While Kittens Game has a "web worker" option to allow it work in inactive tabs, I've had poor results with it. Instead, a work-around I've often used to great effect is to run the game in a separate window. In Firefox, this can be easily done by right clicking the tab and choosing Move to New Window. Other browsers should work very similarly.