Gore won New Mexico by 366 votes. The GOP was ripe and ready to raise legal challenges and push for recounts and whatnot in that state, but the Bush team opted to just put all of the focus on Florida because there wouldn't be much of a difference of winning 271-267 and 276-262
Here, winning Florida and New Mexico would be important.
There's also other states where the margin was really really close. Gore won Wisconsin by 5,708 votes (0.22%). Gore won Iowa by 4,144 votes (0.31%). Gore won Oregon by 6,765 votes (0.44%). I imagine the GOP wouldn't have issue expanding the legal battleground to three other really close states. Meanwhile in this scenario you're positing that Gore barely wins New Hampshire, so I guess that makes it four other really close states.
It'd be veeeeery messy, very drawn out, and a constitutional-political nightmare. Does the House pick the President, given how nobody's technically got 270 when the EC meets? If the situation isn't resolved by January 20th, does that make Dennis Hastert acting President? Can somebody be both Acting President and Speaker at the same time?
Also the reason that it was 271-266 OTL (notice how those two numbers add up to 537, not 538) is because a DC Elector refused to vote as part of a protest of lack of congressional representation. If it's just the OTL map with NH swapped, then it'd be 270-267 Gore-Bush.
----------
With regards to Gore as President, it isn't as if the Clinton administration wasn't bombing the crap out of Iraq for a decade. Gore made a name for himself by being one of the Democrats' Hawks, he was already for removing Saddam from power, and he's got VP
Joseph Lieberman urging him on. If there's a big difference between Bush and Gore, it'd likely be (1) Gore probably won't have a SecDef who handwaves away the Pentagon saying that it's gonna take a lot more than 100,000 troops to occupy Iraq; (2) Gore may not be as dismissive of Iran's offers of rapprochement in the early 2000s the way Bush was; (3) Gore won't be as hands-off on Foreign Policy as Bush was - Bush being more deferential to other more experienced cabinet members, to his detriment; (4) Gore won't have as much of a go-it-alone attitude on the conflict as Bush did; (5) No Cheney giving sweetheart deals to Haliburton and other contractors. More men, more expertise, more international support, more focus, and perhaps less Iranian interference would mean that Iraq would be a very different war from the clusterf**k that it was OTL.
With a good mid-2000s economy, I don't think it's implausible that Gore gets reelected.