In the Android app "AquaMail" I also see older emails in my inbox that are already deleted on the server. My email account is set up as the POP3 account.
Any ideas how AquaMail displays only the actual emails from my POP3 account?
You need to configure the POP3 client and the email service. For example, at Gmail you need to select what to do with read or accessed emails through the POP3 protocol. Most email clients offer you the possibility to select the way to handle the messages retrieved through POP3 or IMAP.