I noticed that your original definition of doing things freely does not apply to all of your examples. For that definition, though, using 勝手 ("one's own convenience/way") may work well.
Don't let your imagination run wild. ("run wild" literally being "in the kind of way that you like/suits you" here.) (Example from tatoeba)
遊び回る does mean" run around," sort of, but more in the sense of doing many positive things. Usually 遊び回る, from what I've seen, has a more positive connotation, as in running around town and having fun, so for your examples it wouldn't make much sense.
As user458 said, 走り回る is a direct translation, but I don't think it would work for any but the first example; I've only seen it mean run around as in run around wildly, like a dog would do; I've never seen it mean something like going to many different places.
For the second example, you may be able to use 走り回る, but I'm not sure. I would think you could use 遊ぶ, as it can mean "hang out" and "hanging out with the hooligans" seems about the same as running around to me. 勝手に遊ぶ may be closer to the meaning you wish.
For the last example, 至る所 (everywhere, all over) might be used to the same effect, if you just replace "run around" with "everywhere," it means the same thing.
I'm not fluent in Japanese, so I'm a bit unsure about the connotations (I'm only judging from what I've seen), so I hope that a native or someone fluent comes to help you, but if they don't, I hope my answer helped!