What is the reason for the different forms of the letter Meem (م) and (مـ) at the end of a word in the Noble Qur’an, and is there a specific rule for this? Also, what is the reason for the different spellings of the word "Ibrahim" between (إبراهيم) and (إبراهم), and do they differ in pronunciation?

There is no objection to writing the letter 'M' in any calligraphic style. The reason for writing the name Ibrahim without a 'ya' (ي) is that the Companions wrote the Qur'anic manuscripts in the well-known style of orthography to accommodate different readings. The word was written without the 'ya' in the Mus'haf of Sham and Medina, and specifically in the Mus'hafs of Iraq for Surat Al-Baqarah. As for the difference in recitation, it goes back to transmission (riwaya), as some reciters read with a fatha on the 'ha' and an alif extension in certain places.