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These are great questions, and everyone's probably going to give a slightly different answer.
I've really enjoyed using Vue.js myself and know that the Laravel community has too. I've done several videos on this which you can see here: https://gorails.com/series/using-vuejs-with-rails (there might be some others not in this series).
A new series I'm starting shortly is going to be building a Trello clone with Vue.js which might answer some of these questions for you too. That should begin in the next week or two.
There is a Rails based Trello clone available on GitHub https://github.com/somlor/kanban .
Thanks! I will check it out.