I'll answer in terms of my stories (not exactly FanFics). I've written both types of chases, so... ^__^

me too, depends of the character, who is more outgoing or shy

For me, it usually depends on who pissed off the other character, surprised them, caught their attention... that kind of thing. xD

Depends on the fic.

It depends on the characters and their personalities, on how they got together in canon if they are a canon couple, and sometimes just on my ideas for that fic.