Did the Messenger of Allah (peace be upon him) and the Companions pray the two Rak'ahs of Fajr after Fajr, and the two Rak'ahs after Dhuhr only, without the two Rak'ahs before it? And is this permissible, or is it considered an innovation?

The time for the Fajr Sunnah (al-Raghibah) is between the appearance of true dawn and the Fajr prayer. It is not permissible to intentionally delay it until after the congregational prayer. However, if a Muslim does so, he does not incur sin, but he has committed something disliked and missed out on much good. It is recommended to make up for it after the obligatory prayer or after sunrise.

Similarly, the regular Sunnah prayers before and after Dhuhr are also Sunnah. There is no sin upon one who omits them, but one should not intentionally abandon them. And whoever misses them is recommended to make them up. The Prophet, peace and blessings be upon him, made up the Sunnah of Dhuhr after Asr. He used to consistently observe ten rak'ahs while at home: two rak'ahs before Dhuhr and two after it, two rak'ahs after Maghrib, two rak'ahs after Isha, and two rak'ahs before the Fajr prayer.