Allah the Almighty addressed to prophet Adam(aw): "Did I not forbid you that tree and tell you: Verily Satan is an open enemy unto you?" They said: "Our Lord! We have wronged ourselves. If You forgive us not, and bestow not upon us Your Mercy, we shall certainly be of the losers." Allah said; "Get down, one of you an enemy to the other (Adam, Eve, and Satan etc). On earth will be a dwelling place for you and an enjoyment, for a time." He said: "Therein you shall live and therein you shall die, and from it you shall be brought out (resurrected)." (7:22-25).

Did Allah forgive him and then send them to The earth? or After saying that dua they had been sent to the earth and then after the incident about doing dua with the name of Prophet mohammad(sw) Allah forgive him?

And after all the hadith some says its mawdo and some says its hasan.

Muhammad (Sallalahu Alaihi Wasallam)said, "After Adam (Alaihis Salam) happened to commit the wrong,(as a result of which he was transferred from Paradise to Earth) he used to spend all his time in weeping, praying and repenting, and once he looked up towards the Heaven and supplicated, "O Allah! I beg thy forgiveness in the name of Muhammad (Sallalahu Alaihi Wasallam)." "Who is Muhammad?" came the enquiry through divine revelation. He replied, "When you created me, I saw the words LAA ILAHA ILLALLAH MUHAMMADUR RASULULLAH written on your 'Arsh', and since then I believed that no human being is superior to Muhammad (Sallalahu Alaihi Wasallam), whose name appeared along with yours." In reply it was revealed," He is to be the last of all prophets, and will be your descendant. If he were not created, you also would not have been created."