The answer regarding 'Umar and his son Ibn 'Umar, along with a group of the Companions, performing 'Umrah in Rajab

The Shaykh : It remains to be said that it is established from 'Umar ibn al-Khattab, his son 'Abdullah, and a group of the Companions that they performed 'Umrah in the month of Rajab. We have been commanded to follow the Rightly Guided Caliphs, among whom is 'Umar ibn al-Khattab. It is said that 'Umar ibn al-Khattab (may Allah be pleased with him) intended that the Muslims not be distant from the House, meaning that if people were told that there is only Hajj in its months, the House would remain almost deserted, especially in the early times before these means of transportation became abundant. People could not come to Makkah except with great difficulty. So 'Umar (may Allah be pleased with him) wanted the connection of the Muslims to what? To the Sacred House. He said, "Perform Hajj in the months of Hajj and perform 'Umrah in Rajab, for Rajab is the middle of the year. Do you agree with this? The month of Dhu al-Hijjah contains the Hajj. Good. Then Muharram, Safar, Rabi' al-Awwal, Rabi' al-Thani, Jumada al-Ula, Jumada al-Akhirah. How many is that? Six. When the half of the year is complete, the month of 'Umrah arrives, so that the connection of the Muslims to the Sacred House is maintained at the end of the year and in the middle of the year. In any case, this is an opinion that came from independent legal reasoning, and a thing is mentioned by its counterpart. In recent times, some people have legislated visiting the Prophet's Mosque.