I'm having trouble deciding which one is correct here. I know that if it just was:
"In this modern era where everyone is busy", Would be correct subject-verb agreement. I'm not sure about the other two below:
"In this modern era, where everyone, adults and children, is busy.."
"In this modern era, where everyone, adults and children, are busy.."
I also have a small question of the usage of "where" here, Should it be where or when?
The subject of be in your sentences is "everyone", so the verb should be inflected for the third-person singular: "everyone, adults and children, is busy...."
"Where" vs. "When" is really a different question entirely but I would say that "where" is fine in this context. Judging by Lambie's comments, some speakers might disagree with me about this though.
I would reword the sentence as follows:
"In an era when everyone, adults and children, is busy..".
Yes, singular verb to match everyone.
I would not use modern, as "in this modern era" is opposed to what? The "modern era" began a long time ago.
Here is an overview of the term modern, including modern era: modern.
I think reading the whole entry clarifies why using modern era is difficult in your sentence.
In this modern era where everyone, adults and children, is busy.
Everyone is singular. Why? Everyone can be "expanded" like this:
In this modern era where every adult and child is busy
Regarding the choice between where and when, the question where wants a place for answer.
How does that relate to time?
Points on lines are places. A common model to talk about time is a "line"--a model so common you have the concept and word timeline.
Points on a line are places and therefore we can say "where does X appear on this line."
Therefore - points of time can also utilize a logical "line" model and questions can use "where".
The question when tends to want things like X o'clock, later, tomorrow, today, something like a specific date as an answer, or a phrase beginning with during.
