In a general sense, an errand is basically an endeavor, usually small in scope, that a person does as part of their overall daily routine. The literal definition is a short journey done to deliver or collect something.
A short journey to carry goods or deliver a message.
i do my errands on time
That is the correct spelling of the plural noun "errands" (tasks, chores).
I'll be running errands for the rest of the day.
no
The expression is not idiomatic. It means exactly what it says. To be sent on ( or for) errands means to be out on a shopping trip, or such like, for someone. Mother sent me on errands to the grocery store and the dry cleaners.
If you "send someone for errands" you're giving them a meaningless task to do just to get them out of your way.
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No. It's not necessary
Of course they did.
The messenger and/or servant
do errands for the rich