Don't like the black hats and frock coats imagery. Not sure how I see that evolving from their traditional garb or fitting their monotheistic, earth-based religion.
You imagery does fit the look of the Paiute Ghost Dance prophet Wovoka (
https://en.wikipedia.org/wiki/Wovoka), but his clothing is European.
If your premise is that North American First Nations built ships and discovered Europe, I picture these explorers and conquerors looking like:
Next you have to decide whether your culture is based on the Mesoamerican model of governance (e.g. Aztec), which is more familiar to old-world European models. Or perhaps the North Eastern style federated government (e.g. Iriquois League's "Law of Peace").
As for their religion, if monotheistic, I would have them worship a Master Spirit. One that can take on male or female form, referred to as "The Great Spirit", a creator God and concneption of universal spiritual force. You may go further and instead of translating as "The Great Spirit" , use "The Great Mystery." But perhaps, while only one god, they still believe that all living things in nature have spirits
Also, there would be no conception of a split between the natural and the supernatural. Therefore the religious memes would be less threatened by science. So even assuming religious fanaticism, perhaps certain sciences advance more quickly than they have historically.
For your purposes I would look more at longhouse religions that place importance in the creators original instructions, which could lead to a an imposed orthodoxy.
You need to come up with a reason why they would want to explore and perhaps conquer the peoples they find in Europe. They would not be driven by lack of agriculture. Are they just an enlightened culture seeking kknowlege? Perhaps upon discovering the europeans, there are goods that Europe offers not available in north america (horses, silk and other fabrics, alcohol, spices). Perhaps there are rival kingdoms/tribes/states in the Americas and while they share a religion and have many cultural simularities, they compete politically and economically. They set up the native american version of the British/Dutch East India Companies and end up colonizing various parts of Europe -- though I would think that they would find India, Africa, and Asia more promising.