How many 5-card hands from a standard 52-card deck contain exactly 1 king and exactly 1 heart? Is my solution correct? 
There are two cases:  
Case 1: (the chosen heart is not a king) So there are $3 \choose 1 $ ways to choose the king, and $12 \choose 1 $ ways to choose the heart, and then $52-12-3 \choose 3$ ways to choose rest of the three cards. 
Case 2: (the chosen heart is a king) So there are $1 \choose 1$ way to choose the king (heart) card, and then since rest of the 4 cards cannot be a king nor heart, there are $52-1-12-3 \choose 4$ ways to choose the rest.
Then adding these two cases together we get the result. 

Almost.
In the first case, once you select a king and a heart that is not the king of hearts, there are $16$ cards from which you cannot select the other three cards: the four kings and the hearts, as $13 + 4 - 1 = 16$.  Therefore, you should have
$$\binom{3}{1}\binom{12}{1}\binom{52 - 13 - 4 + 1}{3}$$
Your second case is fine.
