Well, postfix is a mail tranfser agent (MTA), whereas Dovecot and I guess you mean Courier-IMAP are POP/IMAP servers.
Basically, Postfix worries about listening for new mail (typically on port 25), and then delivers it to a user's mailbox/maildir, a delivery agent (like maildrop), relays it to another mail server on the network for delivery or delivers it to another server on the net (for external delivery).
Dovecot on the other hand sits between a mail application like Evolution, Outllok, etc. and manages classification of mail into new/read, subfolders, deleting mail, downloading (in the case of pop), etc.
The answer to your question depends a lot on how your network and mail applications are set up. For example I have a Postfix server, and use either Outlook or Evolution depending on the OS, but in both of these I explicitly set the name of the mail server, so the MX concept is irrelevant here. In my name servers however, the MX record points to a static IP that allows external mail servers to find the address where Postfix is diligently listening for incoming mail.
If you have port 25 open on your firewall, your main.cf has inet_interfaces set correctly, and your name server MX records point to the correct address, you should be able to receive from anyone.
Don't know about the hotmail question, it seems OK here.
Hope that helps a bit