When an order is approved, does it automatically change my order status to "approved" in Admin?

No. Workflows and Approvals are not connected. One does not know or interact with the other. Any "approval" status in a workflow is meant to be set manually by your staff. 

For example, "we sent a printed proof to the customer and they approved the color." When this happens your internal staff may want to change the order status to "approved" manually. 

In contrast, order approvals in B2B sites where an order is set to require an approval manager first review and approve an order. When the order is approved it results in a check mark displaying in the Approved column in the Pressero Order admin page. No change to the orders status is made.