What Is the First Mosque Built by Prophet Muhammad (peace and blessings of Allah be upon him)?
Question

What is the area of the first mosque that was built by the Prophet (peace and blessings of Allah be upon him)? Was there a mihrab in it? Please quote evidence.
Praise be to Allah.The First Mosque Built by Prophet Muhammad
The first mosque that was built by the Prophet (peace and blessings of Allah be upon him) was the mosque of Quba in al-Madinah al-Munawwarah. 
Ibn al-Qayyim (may Allah have mercy on him) said in Zad al-Ma’ad (3/58), when describing the arrival of the Messenger (peace and blessings of Allah be upon him) in al-Madinah:  
“The Muslims said takbir, rejoicing at his arrival, and they went out to meet him… He continued until he stopped in Quba, among Banu ‘Amr ibn ‘Awf, and he stayed among them for fourteen days, and founded the mosque of Quba, which is the first mosque to be established after the beginning of his Prophethood.”
The great scholar Muhammad al-Amin al-Shanqiti (may Allah have mercy on him) said: 
“With regard to the order of building, al-Masjid al-Haram was the first mosque to be built for mankind. The mosque of Quba was the first mosque built by the Muslims. Al-Masjid al-Haram was built by Ibrahim (peace be upon him) and the mosque of Quba was built by the Seal of the Messengers (peace and blessings be upon him). The site of al-Masjid al-Haram was chosen by Allah and something similar may be said about the site of the mosque of Quba.” (Adwa al-Bayan, 8/326) 
We do not know anything about its area when it was built. It is located south of al-Madinah, but its structure is recent and there is no trace of the first structure. 
The Muslims paid attention to the mosque of Quba throughout the ages. It was renovated by ‘Uthman ibn ‘Affan (may Allah be pleased with him), then by ‘Umar ibn ‘Abd al-‘Aziz. Other caliphs also expanded and renovated it, until the final expansion in 1406 AH. 
The size of the prayer hall alone is now 5035 square meters, and the area occupied by the mosque with all its facilities is 13,500 square meters. 
Virtue of praying in the mosque of Quba
There are hadiths from the Prophet (peace and blessings of Allah be upon him) which speak of the virtue of praying in the mosque of Quba: 
Al-Bukhari (1193) and Muslim (1399) narrated that Ibn ‘Umar (may Allah be pleased with him) said: The Prophet (peace and blessings of Allah be upon him) used to go to the mosque of Quba every Saturday, walking or riding. 
According to a report narrated by Muslim, the Messenger of Allah (peace and blessings of Allah be upon him) used to go to the mosque of Quba walking or riding, and he would pray two rak’ahs there. 
Al-Tirmidhi (324) narrated from Usayd ibn Zuhayr al-Ansari, who was one of the Companions of the Prophet (peace and blessings of Allah be upon him), that the Prophet (peace and blessings of Allah be upon him) said: “Prayer in the mosque of Quba is like ‘Umrah.” (Classed as sahih by al-Albani in Sahih al-Tirmidhi) 
Al-Nasai (699) narrated that Sahl ibn Hunayf said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Whoever goes out to this mosque, the mosque of Quba, and prays therein, will have (a reward) like that of ‘umrah.” (Classed as sahih by al-Albani in Sahih al-Nasai) 
Was there a mihrab in the mosque of Quba?
As for the mihrab, there was no mihrab in it when it was first built. The scholars stated that the building of mihrabs in mosques began in the second century AH. 
For more about mihrabs, please see this answer: 68827 . 
And Allah knows best.