If you have transferred the entire database contents then yes the transaction should show up. If you only did an export via the WordPress admin, that would not have transferred the transactions and addresses tables. The tables are normally named
You can also use the plugin to export the transactions to a csv file and reimport them to the new installation: To transfer the transactions, go to Wallets -> Adapters and choose the “export transactions to .cvs” next to each coin whose transactions you want to transfer. You can then go to the new installation to Wallets -> Transactions and at the end of the page, import the csv file. That should transfer your transactions.
As for IPN history, since you were running the plugin locally, there should be failed attempts listed in your history when you log in to coinpayments. But now that you are running the plugin on a server the IPN requests should succeed.