I was surprised to check my database. There are 28 games with the jump down to G2 in this joseki. Most often (14 cases) Black just plays C3 but there are nearly as many games (12 cases) where Black tests White with the push at G3 first. Pushing forces White to decide whether to pull back or block at H3. If H3, naturally Black plays F2. If White pulls back as in the game, Black turns back to C3 without pushing again.
D5 is a good answer to E3 (which looks like a mistake) but how about E4? This keeps the Black stones connected but it allows White to capture D5. What happens if Black plays C5 (

in the diagram below) and connects D5 instead? If White goes ahead and connects at

to cut, Black will cut in turn at

. Although White can atari at

and capture 4 stones with

, Black ends up with a very big corner and White's shape is over-concentrated. If White approaches the bottom right corner now, Black will not pincer but rather attach underneath and happily take a second corner.
If White simply connects at 4 instead of 3, Black takes 3 in turn and has a much bigger corner than in the game plus the left side is still open. What do you think?
$$Wc Connecting...
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . 0 6 8 . . . . . . . . . . . . . |
$$ | . . . O 4 5 9 . . . . . . . . . . . . |
$$ | . . 2 X 1 X 7 . . . . . . . . . . . . |
$$ | . . X O 3 X O . O , . . . . . , X . . |
$$ | . . X . O X X . . . . . . . . . . . . |
$$ | . . . X . O O O . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
- Click Here To Show Diagram Code
[go]$$Wc Connecting...
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . 0 6 8 . . . . . . . . . . . . . |
$$ | . . . O 4 5 9 . . . . . . . . . . . . |
$$ | . . 2 X 1 X 7 . . . . . . . . . . . . |
$$ | . . X O 3 X O . O , . . . . . , X . . |
$$ | . . X . O X X . . . . . . . . . . . . |
$$ | . . . X . O O O . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]