To join online races, u gotta make gamespy account before start a new profile, by ticking the appropriate button (follow the instructions on screen). Existing off line profile cannot be used online. Yeah I know...it sucks but that's how it is.
Playing over LAN only is not supported, but u can play together with other computers on ur LAN, as long as each has original DVD and all connected to the Internet via the same router.
Instant races are not essential part of the game. The main goal is to make a progress by unlocking the island roads and challenges scattered throughout the map, winning races, fulfilling tasks, buying new cars and houses and so on...the best part (well at least it was) is online aspect of the game, with free roaming, meeting other players, cruising around together, doing some online races....
The servers are rather deserted and inconsistent right now, but frankly, they've never been working very well anyway. Hopefully the next TDU will solve the connection issues