It is a term in jurisprudence within the conditions of the person who can lead a congregational Prayer to Aadil عادل. Aadil means a pious person who does not deliberately commit sinful acts, and if it happens that he committed a sinful act, he immediately repents and seeks forgiveness and never insists on wrong. Such pious, God fearing, obedient person is called Aadil.
(Just) is a literal linguistic translation of Aadil but it does not include the full meaning of the term Aadil.
Congregation Prayer (Salat Al-Jama'ah) becomes invalid if the person deliberately does Rokoo' or Sojood or any other act of Salah before the Imam of Jamaa'ah. Reciting Tasbeehat or Tashahhud or Salaam before the Imam of Jamaa'ah is Makrouh but it does not invalidate the Jamaa'ah while doing the acts of Salah before the Imam of Jamaa'ah does invalidate the Jamaa'ah but not the Salah. Salah then becomes Furada (individual) when the Jamaa'ah is invalid.
Joining a congregation prayer is allowed in every Rak'ah till before the Imam of Jama'ah raises his head from Rokoo'. If you join during Qiyaam (Standing) or Rokoo', it will be counted for you as a Rak'ah.
We are not allowed to join congregation prayer after Rokoo' except in the last Rak'ah just to get the reward of the congregation prayer but it will not be counted as a Rak'ah.
If you are in hurry and can't wait till Jama'at starts, you can pray individually (Furada) but it should not send a wrong message of disrespect to the Imam of Jama'at or to the participants who are waiting to join the congregational Prayer.
It is not allowed during obligatory Prayer (Salah) to add Rokoo' or Sojood because it will invalidate the Salah. Reciting any of the four verses of obligatory Sajda is not allowed in obligatory Salah. If you do Sajda for reciting or listening to any of the four verses of obligatory Sajda, you Salah becomes invalid and you need to perform the Salah again.
Thank you for your question. One of the conditions for a prayer leader according to the School of Ahl al-Bayt (as) is that they are baligh. If they are not baligh, that prayer cannot be lead by them and so if someone participated in that prayer they will need to repeat it.
May you always be successful
congregational prayers are much more better than praying alone. The reward of praying in congregation is many folds more which can be hundreds or thousands of times.
Congregational prayer has conditions including the conditions of the Imam of Jama'at. Woman can lead women only in prayers and she should stand in the first row with other women and not to stand ahead of them.
The main requirements for the Imam of Jama'ah are:
1. He be well versed in the rulings of fiqh and prayer in particular.
2. A man of piety and righteousness, with inner taqwa, high moral standards, and 'adalah, which means he does not commit regular sin.
3. His pronunciation and recitation is accurate.
4. Other things mentioned in the fiqh books.
If there are numerous people who broadly have these main requirements, then the person who is more learnt, or with more taqwa, or better recitation, is presented as the Imam Jama'ah.
And Allah knows best.
According to the verdict of many high caliber scholars, joining the Jum'a Prayer is enough even if you are late and could not listen to the both Khutba of Jum'a.
You do not need to perform Dhuhr.
It is recommended to pray with Muslims from any sect. As it is a Friday prayer, you need to pray your own Dhuhr prayer before or after it.
Praying with our Muslim brothers from any sect is highly recommended.
If it would be within a reasonable range of time after the entering of the prayer time, then yes you can wait, and you will receive the reward of prayer on time, and also jama'ah prayer.
However, if there will be an extensive delay, then it is better that you pray on time, and receive its abundant reward.
And Allah knows best.
It is permissible to perform congregational Prayers keeping the social distancing required by health experts. That distancing will not invalidate the prayer nor the congregation Jama'ah.