こんばんは！ Wow, you ask some deep questions. I'm sure an entire college lecture could be devoted to each of these ^^
～まま（で） The online dictionary we use
doesn't even give a straight English translation for this. I think the best way I can explain it is to say that everything before it is how things are, like what state they're in. For example, そのまま行っちゃった would be, "And then he just left [just like that, with everything the way it was]." I'm not sure if that makes any sense. So the reason you usually get, "At this rate," would be that it's a lot more concise than, "If things keep going the way they are now without changing..." The Japanese seem to have a lot of single words that have a million English ideas packed into them. If you check the 国語辞典 at that online dictionary, it'll give you seven definitions with examples that explain everything very well. We can translate them and post them in another entry if people are interested.
～っぽい/～風に The latter, as far as our experience, is more like, "In the manner of..." It comes from 風（ふう） meaning "manner" or "way." The way ～っぽい was explained to us is that it means "like," when referring to something that seems like something it is not. For example, it would be very insulting to call a man 男っぽい, because it indicates that you think he's actually a woman. But if you use ～らしい ("like" when referring to something that is acting in a way expected of what it is), you're saying he's manly in a good way. But you might say 男っぽい about a manly woman. 日本っぽい might be used when describing something that acts very Japanese, but is not actually Japanese. Memoirs of a Geisha
, for example, might be 日本っぽい, because it wasn't created by Japanese people.
～うちに We don't remember going over this in any of our classes, so this is all speculation, but all of the examples we can think of for this lead us to believe that it means "before ... is over." For example, 知らないうちに would be "before he finds out," or "before his not-knowing is over." Since うち means "inside," you could think of it as, "inside the time frame of ...". 近いうちに was the first example I thought of, which means "soon," "before long," or "within the 'near' time frame."
Does any of that make sense? We'd have to ask our Japanese pen pal if there's any more clarification on the difference between ～うちに and ～間に, but we're pretty sure ～間に is simply, "while A is taking place (B is taking place)."