A third order Franciscan lives his/her life in the spirit of St Francis.
A third order Carmelite lives his/her life in the spirit of desert fathers on Mt Carmel.
Two different types of spirituality.
A third order Dominican lives his/her life in the spirit of Dominic etc, etc.
A third order gives access of these various modes of life to the lay person. First order is ordained; second order is brothers and nuns.
Yes. Discalced means without shoes, so a discalced Carmelite belongs to that group of Carmelites who theoretically go without shoes, but usually nowadays they wear sandals.
Here: meet some of my Discalced Carmelite (second order) friends.https://www.flowerofcarmel.com/daily-life