Is the prayer valid if the worshipper came running to the mosque, and entered into prayer while breathing rapidly and his heart was pounding fast, and he barely recited Al-Fatihah, stopping during its recitation to catch his breath and then continuing the recitation? And does he have to repeat it?

May Allah increase your eagerness. Do not repeat such an action, for the Prophet, peace and blessings be upon him, clarified how to approach the prayer by saying: "When the iqamah is called for prayer, do not come to it running. Come to it with tranquility. What you catch, pray, and what you miss, complete. For indeed, when one of you intends to pray, he is in prayer."

Abu Zur'ah al-Iraqi said: "This contains an command to approach the prayer by walking, and a prohibition against approaching it by running. It should be done with deliberation and dignity. The apparent meaning is that there is no difference between Jumu'ah and other prayers, nor between fearing the loss of the opening takbir, a rak'ah, or the entire congregation, or not fearing that. This is the view of the majority of scholars."

As for the invalidation of prayer by what you mentioned regarding reciting Al-Fatiha, your prayer is not invalidated by that, and you are not required to repeat it.