what is the advantage of playing 3.Bc4 in the kings gambit?

Why? It's quite fine, actually.

After 2.f4 ef 3.Bc4, I was comparing 3. ... Nf6 and 3. ... Qh4+, difficult to see great difference as for position evaluation, proably the matter of taste, but accurate play of the both sides is required

At least, 3.Bc4 "avoids" g5 idea since 3.Bc4 g5 4.h4 looks quite annoying for black (compare with 3.Nf3).

Don't know ... I have nothing against playing the Kieseritzky gambit as White after 3.Nf3 g5 4.h4 g4 5.Ne5. Another thing is that it does not look like Black plays 3.(Bc4) g5 as often as 3.(Nf3) g5

 Another idea of "avoiding" or "annoying" after g7-g5?! on which I have been working is 3.Nf3 d6 4.Qe2 in the Fischer defense. In several blitz games, after 4.Qe2 g5 5.Qb5+ ... 6.Qxg5, for some strange reasons, Black resigned at once, in totally equal position 

 what's the point of Qe2 if black simply develops with Nc6??

I have been working on several responses which I faced - 4. ... a6, 4. ... c6, 4. ... Nc6, 4. ... Bg4, 4. ... g5, and even 4. ... c5 and 4. ... Ne7 (Be7). The point is that in the usual position after 3.Nf3 d6 4.d4 g5 5.h4 g4 6.Ng1 Bh6 (or 6. ... Qf6, with possible transposition of the move order) I strongly prefer to play as Black  Of course, White has also 4.Bc4, 4.Nc3, even 4.d3, 4.h4 or 4.b3. Just from my impressions from 4.Qe2 - it is to my taste, along with the fact that it is less popular and less studied. Namely as for 4.Qe2 Nc6 - I like more 4.Qe2 Nc6 5.d4 (with possible 5. ... g5 6.d5, although 5. ... Bg4 is played more often) than the mentioned 4.d4 g5 5.h4 g4 6.Ng1 (of course, this does not lose for White, but White's prospects and hopes in this position seem to be overestimated, at least in some reference books on the chess openings. And simply from impressions from my games, I like 4.Qe2 more than 4.d4 or 4.Bc4. The matter is that I began playing 4.Qe2 in about 2009, and just out of curiosity - I worked on the lines 4.Qe2, 4.d4 and 4.Bc4 with Komodo 10 and Stockfish 8 - the queen move does not look worse than the other moves, just with the difference that is it relatively rare - and I notice that during the games, indeed.

Its about time Factor too.

In a way yes. But even if you let a chess player sit down at the table with a chess board on it, set up a position which is unknown to him and give him enough time, say, 2 hours for 40 moves, or even 4 hours for 40 moves - there will be difference in quality of moves compared with the position which he played before or compared with some well-known (or popular) theoretical variation. But when working on chess openings, still it is more important to make sure you get (against a strong engine with time control at least 2 hours/40 moves) playable positions which are to your taste, in which you see that there will be not quite simple tasks to solve for the opponent. The point of practicing openings with strong engines is of course to try to be more or less sure that you do not get worse positions than your opponents, and that helps to significant degree to find interesting plans with sound positional base. I mean, neither Komodo 10 nor Stockfish 8, with the time control 40 moves/40 hours have proved me yet that the idea of 4.Qe2 in the Fischer defense is unsound, just like the idea of 3.f3!? in the Caro-Kann defense, 2.b4!? in the Sicilian defense, and other ideas 

I guess the problem i dont like it is mine then, actually.

We are discussing the disadvantages and advantages of Bc4, I simply said that if you want to avoid the modern, you'll have to avoid playing 2.f4 as a whole.

I personally do not see any serious reason not to play 3.Nf3 only to avoid d7-d5. Although I do not play 3.Bc3 (to be more exact, play it rather seldom in blitz games) - upon 1.e4 as Black I from time to time do play 1. ... e5 (my favorite is the French 1. ... e6), and after 2.f4 I always accept the gambit with 2. ... ef. And in the position after 1.e4 e5 2.f4 ef, the oponents from time to time do play 3.Bc4 (probably in about 3-4 cases of 10, sometimes more, sometimes less). In any case I needed to define for myself what I should do as Black upon 3.Bc4. Formely I automatically played 3. ... Qh4+, but now I play 3. ... Nf6, and from time to time trying other ways, too, just for a change and just to see where it may lead , such as 3. ... Ne7!? (really interesting) 3. ... d6, 3. ... d5 (anyway), 3... c6, and even the famous 3. ... b5 from the game Andersen-Kieseritzky :) As for now, the three of them which I prefer are 3. ... Nf6, 3. ... Ne7 and 3. ... Qh4+

Well, 1.e4 e5 2.f4 exf4 3.Bc4 rather invites, than prevents 3...d5.

White has to take it with the Bishop (3.exd5?! Qh4+ 4.Kf1 Bd6 is pretty close to a clear advantage for Black, as the c4 bishop is dumb), and after 3.Bxd5 Nf6 white has to surrender the bishop pair, or sac the e4 pawn, which is almost certainly an unsound option.

3.Bc4 Qh4+ 4.Kf1 d6(!), as suggested in two recent manuals by Bologan and Ntirlis, is a critical line- Black keeps the pawn and wants to castle queenside fast by ...Be6, ...Nd7 etc. The best way for white to meet this plan is quite unclear.