I was told today that the grass radical, e.g. the top of 花 has a gap in the middle of it, and googling it suggests that it is sometime written like this. Is that strictly in Chinese, is it sloppy, or is it just another way of writing? Is the correct way leaving it connected?
The form of 花 with a gap in the radical making it 4 strokes instead of 3 is called the 旧字体 (old character form) and the one that is used most of the time these days is called 新字体.
Neither is correct or sloppy, they're just two different ways of writing the same character. This is related to the fact that characters in general have been simplified in different ways over the years, starting from when they were drawn in the old days in China (艸 for this specific radical). You can see more about this radical here:
I also use a kanji reference dictionary called 漢語林 that lists the 旧字体 along with each 新字体.
It seems to me that 旧字体 tend to be used more often in names than in any other words.