First off, I am assuming that you are using IMAP to download your Gmail to your Outlook and your iPod. You cannot use POP3 for syncing to multiple clients. When you delete messages in IMAP clients(Outlook or iPad), it removes the message from your Gmail inbox but doesn't permanently delete the message.
In order to delete the message, you must first move it to your Gmail/Trash folder. You can permanently delete the message by deleting it after it has been moved to the Trash folder. If you simply leave the message there, it will remain in your Trash folder for about 30 days before it is automatically emptied and permanently deleted.
The article cited below is titled "How do actions sync in IMAP?" and will address your question and related ones. Also, try using the device's mail app instead of the Google app. If you are still having trouble, then go to Google's help page for Gmail and consult the instructions for setting up IMAP.
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