If a non-Muslim wanted to join a Muslim in his prayer, would it be allowed ? And would it be counted as Salat Jama'a for the Muslim ?
Yes, they can.
But unless they have faith in Allah, their prayer goes waste. If they are praying without believing in Allah their energy is going waste. It is just like another form of gymnastics for them.
Allah says in the the Quran in Surat Al-Mā`ūn 107:4-7
So woe to those who pray
[But] who are heedless of their prayer
Those who make show [of their deeds]
And withhold [simple] assistance.
Allah also says in Surat An-Nisā' (The Women) 4:142
Indeed, the hypocrites [think to] deceive Allah , but He is deceiving them. And when they stand for prayer, they stand lazily, showing [themselves to] the people and not remembering Allah except a little.
Where Allah calls the non-muslims hypocrites.
So it is clearly visible from these two verses that Allah does not pay heed to such prayers , where the person is not praying with his full heart.
Hope that answers your question.
I wonder why a non-Muslim would want to join Muslims in prayer. Maybe Muslims payed him to increase the reward of their prayer as Jama'at.
According to Jafari Fiqh, a non-Muslim can join prayer of Muslims but his prayer is not accepted and is cancelled. He will not be counted as number of people in Jama’at prayer but if he is in the line of Muslims in Jama’at prayer the connection of Muslims in lines is not cut and the prayer of who pray by connection of non-Muslim is accepted.
By Online Answering from Official website of Grand Ayatollah Makarem Shirazi
If they are sincere and seek guidance or answers to questions, of course their prayers can be answered. Many people believe in Allah before they officially become Muslims by taking Shahaada, we should never push people away. The above ayah are talking about "Muslim" hypocrites who only prayed so others would see them pray but did not truly have faith. Obviously the non-Muslim is not trying to show others he is praying, he/she is doing it either for curiosity (and even then the heart can respond positively to prayer) or is seeking answers in faith and in that is humbling him or herself....that is sincerity for which there would be reward inshallah.