Is reciting daruood in front of Prophet Mohammed's peace be upon him grave or every Friday in the mosque permissible?
All Praise to Allah Subhanahu wa Taala and blessings of him be on Prophet Muhammad peace be upon him his family and companions
The Prophet (Sal Allaahu Alaiyhi wa Sallam) said:
“Send blessings upon me for your greeting will reach me no matter where you are.” (Abu Dawood)
So no matter where ever you are as stated in above hadith,if in front of grave or masjid you could recite salawat and send blessings on Rasulullah salAllahu alayhi wasallam as it is very mustahab to do so.There is no prohibition mentioned anywhere that salawat could not be recited in front of his grave but a respectful manner should be followed.
"Undoubtedly, Allah and His Angels send blessings on the Prophet, O you who believe ! Send upon him blessings and salute him fully well in abundance" (Surah al-Ahzab,56).
“Whoever visits me after I die, it is as if he visited me when I was still alive…”(Al-Daaraqutni)
"..And if, when they wronged themselves, they had come to you, [O Muhammad], and asked forgiveness of Allah and the Messenger had asked forgiveness for them, they would have found Allah Accepting of repentance and Merciful.(Quran 4:64)
Under this verse some mufassir have mentioned this incident that Imam Ali Radi Allahu anhu said: "Three days after burying the Prophet (Peace Be Upon Him) The Araabi did come and did throw himself on the grave of the Prophet (Peace Be Upon Him), he took the earth and threw it on his head. He said: "Ya Rasulallah! Peace Be Upon Him You did speak and we did hear, you learned from Allah and we did learn from you. Between those things which Allah did send you, is following: (4:64) I am the one, which is a sinner and now I did came to you, so that you may ask for me." After that a call from the grave did came: "Theres no doubt, you are forgiven!" [Tafsir al-Qurtubi]
Sending salawat on fridays are highly recommended and virtuous
It was narrated from Abu Darda’ that the Messenger of Allah (ﷺ) said:
“Send a great deal of blessing upon me on Fridays, for it is witnessed by the angels. No one sends blessing upon me but his blessing will be presented to me, until he finishes them.” A man said: “Even after death?” He said: “Even after death, for Allah has forbidden the earth to consume the bodies of the Prophets, so the Prophet of Allah is alive and receives provision.’”(Ibn majah)
It was narrated that Ibn ‘Abbas said: “The Messenger of Allah (ﷺ) said: ‘Whoever forgets to send peace and blessings upon me, then he has missed the road to Paradise.’”(ibn majah)
Allah Swt and His Messenger knows best.