Would people use this sentence (I do do lots of exercises during the week) if they want to express that a person does exercises regularly during the week. I know the grammar is correct if I say she does do lots of exercises during the week, and the "does" emphasizes the fact that she do lots of exercises. However, "I do do lots of exercises" doesn't sound good to me. What sentence would you use if you want to express the same meaning?
Repeating "do" is generally used to contrast against a previous negative. For example:
I don't do laundry, but I do do windows.
In this case the first "do" contrasts with the previous "don't". Without that, a single "do" is sufficient:
I do lots of exercises during the week.
If you're wanting to respond to someone insisting you don't exercise during the week, I can see why you'd want to use "do do."
I better way of getting the same point across would be to say, "I actually do exercise during the week."