Q 2: Is it from the articles of faith to believe that Adam was the first prophet as was pointed out by Allah's saying in the Qur'an:
Then Adam received from his Lord Words. And his Lord pardoned him (accepted his repentance).
? What are the clear proofs of the prophethood of Adam?
The first messenger to be sent was Nuh (Noah) as was mentioned in the Hadith of the intercession reported in the Two Sahih (authentic) Books of Hadith (i.e. Al-Bukhari and Muslim).As to Adam, it was said that he was a prophet, and therefore he was the first prophet. This is proved by the Ayah (verse) stated in the question and by the saying of Allah (Exalted be He):
And indeed We made a covenant with Adam before, but he forgot, and We found on his part no firm will-power.
There are many Ayahs that denote Allah's inspiration to him. However, we do not have an explicit and authentic proof that confirms that he is a messenger (peace be upon him).May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.