Can we say prayers for the sick of non-Jews during the Amida? If so, do we just say their names or do we say their first name - ben/bat and mothers name. Also, do we finish by saying among all the sick of the children of Israel or when saying the prayers for the non-jew, do we leave that off. I was told we could not change the structure of the prayer, so wondering if I should say prayers for them during the reading of Tehillim instead. Thank you for your time.

Yes, it’s better to pray for gentiles after reciting Tehilim, and you don’t add “among the rest of the sick of Israel”, unless you are also praying for sick Jews, as well. Say their name and their mother’s name, just like regarding Jews.