No, not really. You should have a good understanding of 3D coordinate systems, but the point of using an engine is that most of the difficult math and programming has already been done for you. At this point in time, there's really very little point re-inventing the wheel unless you're either a masochist or you're into engine programming for its own sake.
Quaternions aren't non-linear motion paths; they are complex numbers that are used in this case for storing rotational data, rather than euler angles (this has advantages such as avoiding gimbal lock). But again, you don't have to understand them...the engine does that for you; all you generally do is call a rotate command and not worry about it.
Which is not to say that a good math understanding is useless. It would help you have a better knowledge of what's going on behind the scenes, and what the limitations are, not to mention you might end up writing your own routines or extensions for things the engine doesn't support out of the box. However, saying that calculus is "necessary" is massively overstating the issue, unless, as I mentioned, you actually want to go into the game engine development business, but from what the OP has stated of his intentions, that's way off from his intentions.
Personally, I'd recommend a decent understanding of basic algebra as a good baseline.
Untrue. It's primarily engine programmers who need to know C++. Level scripters, designers, animators, sound people, etc., none of them use C++. Most casual and indie games are made with a 3rd-party or standard in-house engine, which often involves no C whatsoever.
--Eric