As to the risk from e4 being unprotected, what can Black really do about it in the short term? One immediate attack on it which Black can launch is 1...Nf6 (Alekhine's Defense), when White's pawn simply turns the tables on the knight by enacting her "offensive" threat 2.e5, after which 2...Nd5 3.d4 seems to be the main line. The only other immediate attacks on e4 are with pawns themselves, 2...d5 (Scandinavian Defense) and 2...f5 (Fred Defense), both of which White simply answers by capturing the attacker. The only 2-move attacks by pieces available to Black (ignoring the suicidal 1...d6 2...Bf4) are 1...Na6 (Lemming Defense) intending 2...Nc5, which White preempts easily and obviously with 2.d4, and 1...b6 (Owen's Defense) preparing 2...Bb7, but White has time to defend comfortably: 1...b6 2.d4 Bb7 3.Bd3 (however note that 3...Nf6 can no longer be answered by 4.e5, because of 4...Bxg2).
Petroff Defense (2.Nf3 Nf6) is still viable for Black.
The other "good" Black response to 1.e4 is 1...c5, the Sicilian Defense, which sort of sidesteps all the issues raised by 1.e4. Except for leaving e5 as an option for White of course.
So what are the advantages? How can Black get away with this? What is the idea? First, although attacked itself, f4 also attacks Black's undefended e5, so forces Black to make a decision. Second, f4 is latently protected by Bc1; all White needs to do is move the d pawn. So Black's opportunity to grab the f4 pawn outright may be limited; and after grabbing the f4 pawn Black may quickly face the challenge of having to defend f4 herself or lose it to a good White development move (Bxf4). Third, if Black takes on f4, she gives up both the block of the e4 pawn and the control of the d4 square, giving White the moves d4 and e5 free and clear. This third purpose is called "distraction", in this case of the pawn on e5.
Is it worth it? I don't think so, which is why I don't play the King's Gambit as White and try to disprove its effectiveness as Black. Here's one demonstration that the threat against the undefended e5 is utterly nullified by the e1-to-h4 weakness: if Black declines the gambit and even leaves e5 undefended by playing 2...Bc5, White cannot even take e5 due to 3.fxe5 Qh4+; since 4.Ke2 Qxe4 is checkmate, the only choice is 4.g3 when 4...Qxe4+ 5.Qe2 Qxh1 leaves White down a whole rook for nothing.
There are three basic ways for White to meet the main threat, 1) take control of h4 thus directly preventing Qh4+, 2) make a safe(r) nest at f1 for the king to retreat into by moving the Bf1, or 3) shore up g3 with any piece other than the king, so that the g3 defense again works. 3.Nf3 seems the natural way to accomplish #1 and is the best move overall in my opinion, so it is addressed below as the main line. But it is worth considering the other ways to respond and how/why they are not as good.
There are actually four additional ways to accomplish #1: 1a) The immediate 3.g3 of course controls h4 before the queen can get there, but Black being already a pawn up is happy to force pawn trades while further breaking up White's king's defenses with 3...fxg3 4.hxg3 d5; 1b) 3.h4 occupies and controls h4 but is too easily doubled-up on via 3...Be7 4.Nf3 Nf6 leaving both g3 and g4 very vulnerable, to e.g. N-h5-g3 and/or a later Bg4, and leaves f4 untouched; 1c) 3.Qg4, the only move that both controls h4 and shores up g3, with a side benefit of attacking f4, but it fails to attacks upon it followed by Black's second tactic (pin and win of the e pawn) via 3...d5 4.Qxf4 Bd6 5.e5 Qe7 6.d4 f6; 1d) 3.Qh5 also controls h4 with a sneaky Qe5+ threat, but Black just prevents the check and then drives the queen away with 3...Nc6 4.Nf3 Nf6 and the e4 pawn also falls; if instead 4.Bc4 (mate threat), then 4...Qe7 (also pinning e4) with Nd4 and Nf6 to follow.
White has two reasonable ways to accomplish #2: 2a) 3.Bc4, the King's Bishop's Gambit, favored by some over 3.Nf3, but I think 3...Qh4+ 4.Kf1 Nf6 5.Nf3 Qh6 is a refutation (see my blog post); 2b) 3.Be2 is interesting, an only slightly more cramped version of the King's Bishop's Gambit: 3...d5 4.exd5 Qh4+ 5.Kf1 Bd6.
White has three ways to accomplish #3: 3a) 3.Qg4 already discussed as #1c above; 3b) 3.Qf3 which seems to be better than its reputation, both nullifying the immediate threat and attacking f4. One key is that Qf3 is unpinning h2 after g3 is played, because it then covers the Rh1. Black has to find (or know) that the best exploit has now changed to 3...Nc6, having some threats of hitting the queen via Nd4 or Ne5 but more importantly eyeing the fork on c2 (via Nd4 or Nb4) which the queen has abandoned. Book theory then seems to be 4.c3 Ne5 5.Qxf4 Bd6 6.Qe3 Ng4 7.Qh3, but a significant improvement for Black is 4.c3 Qh4+! 5.g3 fxg3 6.hxg3 Qe7, because White's Ng1 is now blocked from defending the pinned e4. So that theory seems to be incorrect and the main line is instead 3.Qf3 Nc6 4.Ne2!, which covers Nd4, attacks f4 a second time, shores up g3, and blocks the e-file pin all at once. (Black Nb4 is met by Na3.) 3c) The immediate 3.Ne2 is bad. Although it shores up g3, 3...Qh4+ 4.g3 fxg3 5.Nxg3 leaves White just a pawn down with a pinned knight on g3.
my blog post. Fischer called this "a high-class waiting move", but there is more to it than just that. The "waiting" part undoubtedly has to do with not rushing to defend f4 when it is not yet attacked. And as Fischer developed the defense in response to losing to a Kieseritzky Gambit (http://www.chessgames.com), it is safe to assume its main purpose is to preempt any Ne5 ideas by White. Other advantages to note: it also stops the pawn from advancing to e5 unhindered, and it releases the Bc8 (to 5 additional squares from 0) (and the Ke8 (to 1 additional square from 1) and the Qd8 (to 1 additional square from 4)). An additional subtle advantage is that it also permits Black the option of d5 (in addition to dxe5) if White plays e5, without the possibility of being captured en passant. Some disadvantages to note: it has blocked the Bf8 down to a single available move (a loss of 4 from the previous 5), and it has opened the a4-e8 diagonal into Black's king. Another possible "disadvantage" could be that it doesn't attack anything the way d5 or f5 would (e4), thus giving White a choice of moves.
More to come in Part 2, including a possible "novelty" for Black on move 7.