Is it permissible for non-Muslims to pray with Muslims in the same row?

It is not permissible for a non-Muslim to pray. Rather, he is first required to embrace Islam, then purify himself, and then fulfill the remaining conditions. A non-Muslim is addressed by the command to pray and will be punished for not performing it if he does not embrace Islam and pray. However, his prayer will not be accepted from him except by embracing Islam. Allah the Almighty says: (And what prevented their contributions from being accepted from them is only that they disbelieved in Allah and in His Messenger) [At-Tawbah/54], and He said: (If you associate others with Allah, then [all] your deeds will surely become worthless, and you will certainly be among the losers) [Az-Zumar/65].

Thus, Islam is a fundamental condition for the validity and acceptance of acts of worship.