"Bullied into" suggests a reluctance to do something, and eventually doing it only because of the bullying. In terms of grammatical function, the action is the result of bullying.
"Bullied for" suggests that the action took place prior to the bullying. Grammatically (though probably not logically or ethically), the action was the cause of the bullying.
As for "run into" vs "met", the former suggests a chance encounter whereas the latter is more general. You can deliberately arrange to "meet" but we would not normally describe such a meeting using the words "run into".
The basic idea for “into” is the same in both cases: some kind of physical or metaphoric entry. “Bullied into something” suggests that bullying (metaphorically) pushed the victim into the situation where they felt compelled to do what the bully wanted. “Ran into” suggests entering someone’s personal space, with the more literal connotation of charging in and knocking them down, or the less literal and less violent sense of accidentally encountering them.