Are Family Gatherings on the Prophet's Birthday and ‘Ashura Permissible?
Question

Is it permissible to have family gatherings – among brothers and cousins – and to eat together on special occasions and ‘Eids (by special occasions I mean the Prophet’s birthday, ‘Ashura etc)? What is the ruling on one who does that? And what about gatherings after someone had memorized or completed the Quran?
Praise be to Allah.Exchanging visits and getting together
Undoubtedly exchanging visits and getting together with brothers, cousins and relatives on the ‘Eids that are prescribed in Islam (namely ‘Eid al-Fitr and ‘Eid al-Adha), and on happy occasions, is something that brings happiness and increases love and strengthens the bonds between relatives. 
But what happens in many of these family gatherings, such as mixing between (non-mahram) men and women, even if they are relatives and cousins, are bad customs that go against the teachings of the Quran and Sunnah, which enjoin lowering the gaze and forbid wanton display, being alone with a member of the opposite sex (khalwah), shaking hands with non-mahram women, and all the things that lead to fitnah. 
The Prophet (peace and blessings of Allah be upon him) warned against the dangers of taking matters lightly with relatives. He said: “Beware of entering upon women.” A man from among the Ansar said: “O Messenger of Allah, what do you think about the in-law?” He said: “The in-law is death.” (Narrated by al-Bukhari, 4934; Muslim, 2172.)
Al-Layth ibn Sa’d said: The in-law is the brother and similar relatives of the husband, his cousin and so on. It was also narrated by Muslim. (Please see question no. 1200  for more discussion on the issue of mixing). 
Celebrating the birthday of the Prophet and ‘Ashura
With regard to celebrating the birthday of the Prophet (peace and blessings of Allah be upon him), ‘Ashura and other occasions, and taking them as special occasions and festivals (‘Eids), we have explained previously that there are only two ‘Eids or festivals in Islam, namely ‘Eid al-Fitr and ‘Eid al-Adha, as the Prophet (peace and blessings of Allah be upon him) stated. Please see questions no. , 10070  and 13810 . For the ruling on celebrating ‘Ashura’ please see question no. 4033 . 
Gathering to congratulate one who has memorized the entire Quran
With regard to the family expressing happiness and gathering to congratulate one who has memorized the entire Quran I cannot see anything wrong with it in sha Allah. This is a recently invented kind of celebration, but it is okay so long as they do not make this day like an ‘Eid which they celebrate every year. 
It becomes clearer that this celebration is acceptable if the one who has memorized the Quran is young in age, and needs to be encouraged to continue checking it and helped so that he will not neglect it and forget it. 
And Allah knows best.