Is it bad if all my kittens starve?

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?

Your kittens gain experience at their jobs. This is only visible after you have unlocked the technology Civil Servant at your Library. A list of your kittens and their experience levels then becomes visible in a Census div in the Village tab.

This means that the longer a kitten maintains a certain job, the better they become at it. If your kittens die each year, or if you change their role frequently, you’ll lose some efficiency.

The only long term downside (short term being not having workers to generate resources and having to reassign them after) is losing the ability to gain the achievement "Serenity" for having 50 kittens without ever losing any. I believe the chance to get this achievement is available again after resetting the game though.

Also, when your kittens die, they lose their profession mastery and the new ones will work slower.