How can I add in a 60 seconds countdown timer?

You can use this expression over the Source Text parameter of your text layer:

linear (time, 0, 60, 60,0).toFixed(X)

Your composition must be at least 60 seconds in duration. You can change the "X" in the expression for the number of decimals you want to use.

This way you can easily change the style and format of your text using the Character panel.

The challenge with uisng an Effect like Numbers and Timecode is that you don't have any control over the options that appear in the Character panel and the Paragraph panel.  

I've done some countdowns by keyframing the Text Source parameter.  It's not nearly as fast as using one of the Text Effects, but gives you full control of the appearance.  Another slow down is that changing just one option in the Chracter panal later needs to be done at each keyframe.  On the upside, it opens up all of the options available in the Animatte pop-up menu.

"Timecode" effect can be useful animating the "Starting Frame" option. But with "Numbers" you have best results.

Create a Solid and then apply the effect Numbers located in the Effects > Text category. You can animate the numbers to create the countdown.