"In God we trust" has an archaic (an old-fashioned) style to it. But that also makes it sound kind of solemn and/or dramatic and important. I've played a bunch of Japanese video games whose English localization uses old-fashioned language such as "doth thou wish to find the Master Sword?" (I'm making that up, but you get the idea) and that has a similar effect. It's not how people talk today, but it has a certain style and it conveys a certain mood. You're right, US banknotes could say "We trust in God" and it would mean the same thing and it would be more like how people talk today. Here's some history on the motto: https://www.treasury.gov/about/education/Pages/in-god-we-trust.aspx. Now for "I trust you" versus "I trust in you" versus "in you I trust". "I trust you" generally means that I believe you're not going to lie, harm me, nor let me down. "I trust in you" is a deeper and less personal sentiment, which is why you'd say "We trust in God" and not "We trust God". "We trust God" would be appropriate if we'd just loaned Him a few dollars and we believed we'd get it back when he said we would. "We trust in God" is more along the lines of trusting him to be great, to lead us, and to look after the Universe. "Trusting in" is a bigger, and more solemn, concept than just "trust". And, finally, if you wanted to sound archaic as well as deep you'd say "in you I trust". :)

"In God we trust" emphasizing "In God" since it's a motto or somehow like a proverb it is an exception. You wouldn't normally use this in normal conversations. There's a history to that https://en.wikipedia.org/wiki/In_God_We_Trust .

It's just an inversion to stress the sense of the frase