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.
That's exactly the situation I was in. Yes, you can use Fallout 3 savegames (boxed) with Fallout 3 GOTY (steam) (if you had any mods installed in your original install, it will tell you "Some features of this savegame are no longer available. These features will be disabled." - my savegames still worked perfectly though)
I had to move the save files to a slightly different location, though. My original saves were stored in
C:\Users\BlueRaja\Documents\My Games\Fallout3\Saves\
(The XP-equivalent of that would be C:\Documents and Settings\BlueRaja\My Documents\My Games\Fallout3\Saves\)
My new saves were stored in (and thus my old saves had to be moved to):
C:\Users\BlueRaja\Documents\My Games\Fallout3\Saves\BlueRaja\
I would recommend installing and starting the new copy and, if it doesn't recognize your old saves, create a new game and save it real quick to let it create the save-directory, then move your save files to the new directory.
Best Answer
Take a look at Fallout Wanderer's Edition. It's probably the best regarded overhaul for the game, and should be tweakable to get whatever you happen to define as 'playing like an FPS'. If FWE doesn't float your boat, the Nexus is home to other overhauls that might suit your needs better, but FWE is the big one, and it's the one I'm familiar with.
Oh, and don't use VATS, but that goes without saying.