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I have only ever been able to find out when Moz crawled the site by finding the information in the crawl report. If a crawl report was generated then the information should be on the report. If you can't see it on the site, download the CSV file and it will appear within the first 5 rows of the spreadsheet. If no crawl report was generated and you have a campaign set up, the information will also be within the campaign inside of Moz Pro.

It's not uncommon to see some lower traffic as Google re-crawls the site. If your 301 redirects are all done correctly to the correct pages, there's not much else you can do. As James mentioned, you may want to reach out where appropriate and see if you can have the link directed to the new URL. I don't know how much they've changed, but you will also want to check and make sure that any changes to the internal links or navigation are for the best. Deeper pages can sometimes suffer in site redesigns if the number of internal links or the click-depth has fallen.