Is it permissible to celebrate the birth of the Prophet and to praise and extol him without exaggeration, and how should he be praised, and what is the explanation of the hadith: "Do not exaggerate in praising me as the Christians exaggerated in praising Isa the son of Maryam"?

Celebrating the Prophet's birthday is impermissible because it is an innovation (bid'ah). An innovation is any matter that the Prophet, peace be upon him, refrained from doing despite the presence of a cause for it and the absence of any preventing factor. There is no religious legislation for celebrating his birthday. As for praising the Prophet, peace be upon him, with attributes that do not constitute blameworthy flattery, such as praising his good character and conduct, there is no harm in it. The prohibited flattery is exaggeration in his praise, by praising him with attributes that are exclusive to divinity, such as knowing the unseen or controlling the universe.