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But for the question 1, do you have limited turn to do so or can you raise the traits (Army Quality) until you feel it is enough without worrying about turn, days, budgets or side events?

The game does not include grinding in general. That said, the passage of time isn't really a management factor either. Rather, time passes when you advance the story, so you have as much time as you need to find all the events available in a given section before moving on. So, while there are no unlimited sources of variables, the game is also balanced around the number available before that point (and, on balance, I prefer to err on the side of avoiding frustration).