If your old address was linked with your existing Apple ID, there should be no problem since you will not need to validate the account. However, if it was not previously linked to your current Apple ID, you will need to contact Apple for help resetting the password as previously stated.
You are unable to remove one of your iPhone’s email accounts. If the email account is a business account, it may have a profile on your device. Go to Settings > General > Profile and then “Delete” profile to get rid of the profile.
You can update your Apple ID on iOS without access to the original email by using the same technique on iTunes. Luckily, you can still log in to your Apple ID at appleid.apple.com as long as you can still log in, thus updating the Apple ID login to that new address.
Yes, you can recover deleted emails from your iPhone permanently. If you delete an email from your iPhone’s inbox, it will be deleted from your server. However, if you delete an email from your iPhone’s trash, it will be on the server. To recover a permanently deleted email from your iPhone, open the Trash and pick the email you need to restore. Then, tick the Move button on the upper-right side of the screen and choose inbox.
To recover a complete email from iCloud, open the email in your iCloud account. Next, tick on the “more” button at the top of the email and pick “Download.”