As for sending, the email server makes the outbound connections to where ever it needs to to deliver the email, eg other ISP's, companies etc. It can be configured to send everything through your ISP's email server, but that's not necessary and considerably less geek-worthy.
It's worth noting that many ISPs (and RBLs etc) will drop (or at least weight highly negatively) any mail originating from an IP that is a registered dynamic block. You might find that running your own MTA on whatever IP your ISP gives you works out ok. Or you might find that some mail gets dropped completely by the receiving MTA, with or without notification. There has also been increasing noise for ISPs to drop any port 25 traffic originating from their dialup/dsl networks, as the vast majority of it is going to be spam. Personally, I wouldn't risk it :) I believe this "correct" solution in this case is for your home LAN email server to smarthost through your ISP's MTA, or through a colocated MTA you or a friendly person controls, making good use of TLS and SMTP auth. It maybe not be as geek-worthy, but it's more robust.