re: Is there a certain age people starting counting down the years they have left?

Not exactly, but once you start getting to middle age, you likely have a certain amount of friends who have died, or watched stars/celebrities you grew up watching die. It starts to creep in that life has an end. Problem is, you don't know when that end is.
At this point you have a number of approaches you can take:
1: live every day like there's no tomorrow, don't fear the reaper
2. freak out about your health, become a health and fitness buff if you weren't already, stop smoking, drinking, doing drugs, eat only lettuce, lean meat and broccoli OR become vegan, OR follow some other weird diet regimen that you annoy all your friends about, start taking a lot of supplements, panic over every number your doctor tells you is sub-ideal
3. Take a sort of moderate approach and do the obvious things to improve your health but understand a certain amount of your future is up to fate and you can be in good health but get hit by a bus tomorrow and try not to worry about it too much.

I can't think of a more ironic waste of time than doing this.