How do I hide my default Gmail email address in the compose or reply drop-down menu?
You need to add an additional account (ie. email address) from which to send emails in order to change the FROM email address when you compose or reply to emails. Once you have specified additional email addresses from which to "Send mail as" then you can specify any one of these as the default.
You can also specify whether to (by default) always reply from the address that the email was originally addressed to, or always use your default email address (whichever email address you have specified to be the default one).
Settings > Accounts and Import > Send mail as [Send mail from another address]
In order to add additional email addresses, you need to first prove that you already have access to that account by following some instructions in the confirmation email sent to that address.