I finished Fallout 3 and was satisfied with the story and the way it ended, am I missing something if I don't play the multiple expansions that came out afterwards?
Also, will Fallout New Vegas be connected (storywise) to Fallout 3?
No to both questions. The only one which might come close is Broken Steel, which expands on the core story a bit. However, it isn't necessary for any core understanding of the main storyline.
The other advantage of Broken Steel is that it does (SPOILER ALERT) allow you to continue playing after the final quest.
According to the Fallout Wiki, knowledge of Fallout 3 is not needed for Fallout: New Vegas, and there are no storyline connections, although like in previous games, there will be the occasional reference to the prior ones in the series.
(I should note that I've not played the original two, and had no issues with 3 and its references to them, so I'm guessing any references from New Vegas to prior ones will be similarly subtle. I do, however, plan on playing the original two!)
Finally, in terms of the DLC, I did purchase, play and enjoy all of them. I was not super-thrilled with Operation: Anchorage, but I thought the others were all exceptionally well-done (even if Mothership Zeta had really nothing to do with the Fallout universe). So naturally your mileage may vary, but it's worth noting that if you want some great additional (and different) stories, the add-ons are, to me, worth the money. Broken Steel is the only one which didn't offer a completely new environment: The others all offered large new lands to explore, with stories to go with them.
1 - Not really. The only DLC that extends the main story line is Broken Steel which allows play after the original game ends and continues the fight against the Enclave. It also moves the level cap from 20 to 30.
2 - Fallout: New Vegas takes place 3 years after Fallout 3. The story is not related to the events in Fallout 3.