In your first diagram, maybe try 10.Kf1 because as you noticed it's not at all safe on d1.
In the 2nd diagram, I don't know what you mean compensation, black is down the exchange with nothing to show for it.
Black never plays g6 (unless you're facing a beginner).
Of course, if you're seeing f6, chances are you are playing a beginner, but it doesn't hurt to look at lines where black doens't immediatly kill himself.
What were the time limits on these games? Usually with a response like that you're either playing a beginner or your opponent is trying to put the pressure on you with an unorthodox opening (again, only really works if you've not really got enough time to see your advantage).
I'm not an amazing player by any regard, so I'm sure someone else has a better analysis, but apparently the main line goes like:
But I'm sure fxe5 is very common, probably just common as the above line.
In your first game, maybe Bb3 to take advantage of his weakened diagonal. He's so boxed in he'll probably move his King's Bishop. If Bb4 then c3 and d4. If Bc5 then I'm not sure, probably the same thing. Despite the fact you haven't taken e5, his Kingside safety is severely compromised and your development will be much better. Perhaps the best move would be d4 to transpose into some Scotch variant, perhaps after you've exchanged the Bishop for his Knight. Even after the Queen exchange (which seems pretty forced to me, Black's attack after Qxe4+ seems pretty powerful, enough to reverse advantages even.) your superior development (e.g. d4 for the Classical Centre or 0-0) and you seem to have good compensation for the lost pawn. I think the same holds in the second game, take the Queen and develop your pieces.
I'm sure, however, that someone better then me can produce a better analysis. 
Thanks guys :)
I'm sure most players who play 1. E4 know that it's a bad idea for black to play f6. However, I've come across a few somewhat scary situations against this.
After seeing this, I wonder if 2...f6 is really a "losing" move for black. Not recommended, perhaps... but it appears a fairly strong counterattack can occur if they grab your rook. Obviously, the difference in that line is that black's biship is free, which can make a big difference, but can black gain an advantage in the main line?
What are your thoughts on these two games?