No. The bishop and knight will be far superior to the lone rook and pawn, and the king can be defended.
Is a trade of bishop and knight for a rook and pawn plus exposing the king good in the Italian?
#1
No.
In theory it would be an even trade: B + N = 3 + 3 = 6 and R + P = 5 + 1 = 6.
However, you trade 2 developed pieces for an undeveloped rook and an undeveloped pawn.
You invest 5 moves: Nf3, Bc4, Ne5, Bxf7+, Nxf7
Black only spends 2 moves: Rxf7, Kxf7
Hence you lose 5 - 2 = 3 tempi.
3 Tempi is about equivalent to the loss of 1 pawn.
The loss of 1 pawn is enough to lose the game.

#1
No.
In theory it would be an even trade: B + N = 3 + 3 = 6 and R + P = 5 + 1 = 6.
However, you trade 2 developed pieces for an undeveloped rook and an undeveloped pawn.
You invest 5 moves: Nf3, Bc4, Ne5, Bxf7+, Nxf7
Black only spends 2 moves: Rxf7, Kxf7
Hence you lose 5 - 2 = 3 tempi.
3 Tempi is about equivalent to the loss of 1 pawn.
The loss of 1 pawn is enough to lose the game.
Thanks, so what's a good strategy when my opponent does this and exposes my king.

#1
No.
In theory it would be an even trade: B + N = 3 + 3 = 6 and R + P = 5 + 1 = 6.
However, you trade 2 developed pieces for an undeveloped rook and an undeveloped pawn.
You invest 5 moves: Nf3, Bc4, Ne5, Bxf7+, Nxf7
Black only spends 2 moves: Rxf7, Kxf7
Hence you lose 5 - 2 = 3 tempi.
3 Tempi is about equivalent to the loss of 1 pawn.
The loss of 1 pawn is enough to lose the game.
Thanks, so what's a good strategy when my opponent does this and exposes my king.
If the opponent does this, it means that you already have more activity, if you followed opening principles up until that point and you have lead in development. Develop the rest of your pieces, and just play. At certain point you might want to play Kg8 to make your king more safe, Let's see this example on the board:
One thing that you should realize is that 2 minor pieces for a rook and a pawn means that you have 1 more piece than the opponent in total. This is 1 piece more that can attack 1 single square. So you should try to find weaknesses and aim your pieces at them, because with 1 less piece in the game, the opponent will have a more difficult job defending their weaknesses. There is no immediate win, it is just that black has a better development and a favorable imbalance.
The only situation where 1 rook with a pawn can possibly overpower 2 minor pieces is in the endgame. There are actually examples, that even a rook and 2 pawns are not better than 2 minor pieces, especially if the side with 1 less rook remains with a bishop pair.

Another thing to consider is the B and N are really valued at more than 3, N closer to 3.25 B 3.5 add the tempo described in detail by tygxc and you have given up 7.75 for 6. As explained by others the K on f7 is not really in danger. In different positions a trade like this could be looked at like this a B can control/threaten 14 squares the N 8, on the other side the R 14 and the P 2; 22 vs 16 of course if this trade leads to Mate then go for it, but then it is not a trade in is a combination!

Similar to above. I always tell my chess students if they are giving up the two pieces and taking the rook, then I need *two* pawns to call it even material, not one. Maybe in the ending, where the value of pawns goes up a bit, I’d say R+p vs B+N is close to equal, but in opening and middlegame, especially when rooks are more difficult to activate anyway, forget about it. The two minor pieces are just much much better than a rook. Then we also consider how many moves it took for you to do the trade, like in the post #5 example, and when we add everything up white’s probably lost objectively.
I was just wondering whether that trade is good and if it's OK what possible strategies do I have?