If you are asking about setting up a game and playing head to head or co-op via system link, yes. Xbox live is not needed for that. The game will be hosted and played on the linked consoles. If you want to play other players on xbox live, you need to have xbox live. Linking your systems together will not allow you to bypass that.
halo 3 odst, unlike halo 3, has no matchmaking, sorry
by playing the game and winning matchmaking
Without matchmaking or the Oni Sword Base farming, there's no fast way to do it. Challenges are the only way I can think of and most of those require matchmaking.
To play on Halo 3 matchmaking you must have a gold xbox live membership and an internet connection to play on matchmaking. If you only have a silver membership it will not let you play matchmaking.
General grade four
No not unless you were banned from Xbox live all together.
yes you can, Firefight will however be different in matchmaking then in a custom game. The match will usually be on Heroic difficulty and will only be 1 round. but yes you can play with random people in firefight matchmaking.
It's a rank.
Matchmaking
yup
Yes. Halo 3 and Halo: Reach both have Matchmaking.
Shoot and kill people.