Will AWI process an order that was placed before the days set in number of days to search and approved withing the number of days to search?

Yes, AWI will process the orders those were approved within the number of days to search set in AWI settings irrespective of the time when these orders were placed. Pressero does not generate a production file if the approval plan is active and order is not approved. Once the order is approved, Pressero system will generate the production file and inform the AWI about the order and then AWI process the order as soon as the order is approved (almost real time considering 2-5 minutes delay due to network quality)
This works the same if you are using an integrated payment method and due to any reason the order was paid later. Once the order is marked as Paid, Pressero system will generate the file and inform AWI about the order that AWI will download.
In case of Active Approval plans and Integrated Payment methods AWI will not consider the time of the order is placed but considers the time when it is approved and/or paid.