I don't think this is so much the difference between the two but rather a combination of the two. Here the て-form is used in the continuous sense, 休んでいる, and the なさい is simply applied to the end. In ...
Answer is, no.
Following are not orders, not requests:
あなたたちは頑張るぞ (you guys)
彼らは頑張るぞ (those guys)
私は寝るけどみんなは頑張るぞ (everyone except for me)
Following may ...
nasai and nasaimase are イ音便 forms for nasare and nasarimase.
nasaimashi is an uncommon alternative for nasaimase.
nasai and nasaimase are different in that mase is more formal (for a customer perhaps) ...