Basically a lot of things is optional, but they might make your work faster or more effective. It also depends what are you working on, doing C# in IDE is way lot easier than doing it without, but for C++ I probably would argue as I think that working in C++ is more effective without IDE. All in all it is personal matter, some people prefer IDEs, some don't (it's like "some prefer beer, some wine") - so I can't tell you which IDE you want, or whether you want it at all.
From my experience, using IDEs tends to less thinking about the code, but also that is definiely not rule.
As for API, it depends on whether you're going for Windows, Linux, Cross-platform, etc. If you really want to work from scratch I'd recommend you http://www.sfml-dev.org/, as it is very easy to use and it can be learned in an instant, which is always a good thing (although sometimes in the future you might need to directly work with XLib/Windows/... - learning those might be good experience, but takes a lot more time). Of course using SFML often means using OpenGL and OpenAL as graphics and audio library.
As you wrote that you're short on time, you might just grab OpenGL or D3D (based upon which one you actually learned in school), and start working on game. As for IDE, you can go with mentioned Visual Studio, I'd recommend also Qt Creator or Mono Develop (in case of C#), in case you're Linux user. Unless you don't know a lot about bash scripts, I don't recommend following the way of I-work-withouth-any-IDE ... although in that case you probably wouldn't be asking as here.