Is it permissible to say: "It is forbidden for Allah to do such and such," like an arrogant person entering Paradise? And is this considered a transgression against Allah, or is it similar to the Hadith Qudsi: "Allah has forbidden injustice upon Himself"?

Allah may obligate Himself to whatever He wills and forbid whatever He wills. However, it is not permissible for creation to obligate anything upon Him or to forbid anything, because that is from the saying of the people of innovation. And Allah, Exalted be He, is the مالک (Owner) of all things; He does whatever He wills, and there is no one to overturn His judgment. The scholars have agreed that what is obligated by His truthful promise is what He has written upon Himself, such as: "Your Lord has decreed upon Himself mercy" and "Indeed, I have forbidden injustice upon Myself." And it is not for the servant to obligate anything upon Allah.

As for the hadith: "No one will enter Hell who has a mustard seed's worth of faith in his heart, and no one will enter Paradise who has a mustard seed's worth of arrogance in his heart," it is one of the hadiths of warning. There is a difference of opinion regarding its interpretation, and the verified scholars have concluded that whoever has arrogance in his heart will not enter Paradise initially; rather, he may be punished and then enter it, or Allah may forgive him, because a monotheist will not abide eternally in Hell. This is like other hadiths of warning that do not permanently forbid Paradise to a monotheist. And a Muslim should not be overly elaborate or utter statements for which there is no revealed text.