Suddenly all but 30 of the kittens vanished and none were working

No, it's not really a problem. Assuming you're there to re-assign them.

The big problem is if you make sure you can survive a normal winter (-75% catnip rate), leave the game idle for a while (say overnight), and at some point, a cold winter hits (-90% catnip rate). If a kitten dies while you're idling, then the new kitten to arrive won't be assigned a job until you get back.

There are a few achievements related to starving kittens, though. You need to starve kittens for two of them (one at ten dead kittens, one at 100), and another comes for reaching 50 kittens without any dying. You don't need to worry too much about missing that last one, though; it's easier after a reset or two anyway.

Of course, the real question is: are you going to live with yourself after letting a poor innocent kitten starve to death?

It is currently hardcoded to only show 10 kittens in each job, so you cannot view all kittens if you have more then 10 in a job.

Inside the source you can view the following for-loop:

for (var i = sim.kittens.length - 1; i >= 0 && kittensLimit < 10; i--)

This loop populates (via this.records.push) the visible kittens, and the kittensLimit max sure there is a maximum of 10 kittens shown. In the update-function (same source) the following for-loop can be found:

for (var i = 0; i < this.records.length; i++)

and the pushed kittens are shown on your screen, with all the council links et cetera.