Is it Bidah or Sunnah to let the person on your right side to enter the door?Is it Bidah or Sunnah to shake hands from your right hand side onwards when you are shaking hands with a group of people?

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Entering mosques is a form of obedience in which Allah asks us to compete and strive to be the best. So, at the door of the mosque, the Muslim should not give another Muslim the priority to enter before himself if he was the first to arrive. It is narrated from Sahl Bin Sad that he said: "Once a tumbler (full of milk or water) was brought to Allah's Apostle who drank from it, while on his right side there was sitting a boy who was the youngest of those who were present, and on his left side there were old men. The Prophet asked, "O boy! Do you allow me to give (the drink) to the elder people (first)?" The boy said, "I will not prefer anybody to have my share from you, O Allah's Apostle!" So, he gave it to the boy".
So, any act that can make someone nearer to Allah, he should not favor any other person in doing it as this does not conform with competing in righteousness and good deeds.
As for other acts which are not religious such as coming out of Mosque etc. one may give another Muslim the priority to go out particularly the one who is on his right hand side. But the right to leave first is for the first person who comes to the door.
But whoever greets a group of Muslim and wants to shake hands with them he has to begin from the right side since the Prophet preferred to use his right hand in all actions; (for instance) for performing his ablution, for combing his hair and for putting on his shoes". [Reported by Bukhari and Muslim].

Allah knows best.