The Cold Food Festival is mainly established to commemorate Jie Zitui.
It is said that during the time of Jin State, Prince Chong'er fled to a remote place with his ministers, including Jie Zitui, to avoid turmoil. When Chong'er was hungry, Jie Zitui cut a piece of meat from his own leg, roasted it, and gave it to Chong'er to eat. After Chong'er returned to the capital and became the ruler, he rewarded his loyal ministers, but forgot about Jie Zitui. Someone complained on behalf of Jie Zitui, and Chong'er felt guilty. He sent someone to invite Jie Zitui to accept the reward and an official position. However, Jie Zitui refused and hid with his mother in Mianshan Mountain.
Later, Chong'er personally went to Mianshan Mountain to invite Jie Zitui, but Jie Zitui still did not want to come out. Chong'er then ordered a fire to be set to the mountain, intending to force Jie Zitui to appear. Unfortunately, Jie Zitui and his mother were burned to death under a tree. Chong'er looked at Jie Zitui's body and regretted deeply. When burying him, he found a willow tree hole in his back, which seemed to contain something. Taking it out, it was a piece of clothing with a poem written on it: "Cutting flesh to offer you, my heart is loyal. I only wish your majesty remains clear-minded. If I die beneath the willow, it's better than being a minister who advises you. If your majesty truly remembers me, he should always self-reflect when thinking of me. My soul in the underworld has no guilt, and I hope you govern diligently and remain clear-minded."
In memory of Jie Zitui, Chong'er ordered the name of Mianshan Mountain to be changed to "Jieshan," built a temple there, and designated the day when the fire was set as the Cold Food Festival, where people would not light fires and only eat cold food. The next year, Chong'er led his ministers to climb the mountain in plain clothes to pay homage, expressing his sorrow. After the memorial ceremony, Duke Wen of Jin named the revived willow tree "Qingming Willow" and set this day as the Qingming Festival.

Yes.
In ancient China, the Cold Food Festival was much more important than the Qingming Festival. However, with the development of the times, the two festivals have gradually merged.
However, during the Three Kingdoms period, Cao Cao issued an order to abolish the Cold Food Festival.
On the day of the Cold Food Festival, people did not cook, ate only cold food, and this was just a special custom in the Shanxi region. But Cao Cao thought it was a bad custom, fearing that the people in the cold areas of Shanxi might be frozen or starved, so he issued an order prohibiting it, with the penalty being half a year in prison if violated.