Is it safe to remove mails on imap server with “rm”

A quick google throws up IMAPFilter which looks like it does what you want:

• Searching of messages using many available criteria in the IMAP protocol, such as:
  • Status (recent, unread, etc.) of a message.
  • Size of a message.
  • Age of a message.
  • Matching of a string or a regular expression pattern in the headers or the body of a message.
• Ability to use logical operators (and/or/not) while searching messages.
• Different capabilities available to process messages, including:
  • Deleting messages.
  • Moving messages to a mailbox in the same or different servers
  • Copying messages to a mailbox in the same or different servers.
  • Marking messages or changing message flags.
• Listing of the available and/or subscribed mailboxes, along with support for the IMAP CHILDREN extension.
• Creating, deleting, renaming, subscribing or unsubscribing mailboxes
• Internationalisation (I18N) support.
• Server namespace support using the IMAP NAMESPACE extension.
• Secure Socket Layer (SSL) or Transport Layer Security (TLS) encrypted imaps (port 993) connections.
• Encrypted connections using the IMAP STARTTLS extension.
• User authentication with the Challenge-Response Authentication Mechanism (CRAM), specifically CRAM-MD5.
• Perl Compatible Regular Expressions (PCRE) support.

Here the proposed mail-flow-diagram for your case.

External mail server
   |        ▲
   |        |
  POP3     SMTP
   |        |
   ▼        |
Local mail server

First, you need setup working local mail server with postfix and dovecot. The important parameter for postfix is never list your local domain in relay_domains. You can use virtual_mailbox_domains or mydestination for local domain.

Second, you need to populate/synchronize the user from external mail server to local. So, external mail server has the valid recipient list for receiving email and local mail server has user database for authentication and management.

For outgoing email flow, just setup SMTP submission service with SASL. Then setup relayhost in postfix to your external mail server. You should also setup postfix SMTP client with single username or other method. It would depends on your provider mail server.

For incoming email flow, you should use cron/other daemon method to periodically check INBOX in external mail server, download new messages via POP3 (don't forget to delete it after download) and put it in the right mailbox. Fetchmail will be suitable for this task.