Welcome Erik,
In ressponse:
Is it possible to add your own classes and reuse them across algorithms?
To re-use code, you need to copy-and-paste. There is no way to import your own custom modules, or set up a MATLAB-like search path.
Can I add additional files or tabs to the IDE to better separate my code?
No.
Can i use the functionality that you have developed outside of your website (locally for example) through an api or something along these lines?
Kinda sorta. My understanding is that the Quantopian Research Environment (or whatever it is formally called) will eventually be rolled out to all users as a free offering. It'll be more flexible. There is also zipline (https://github.com/quantopian/zipline), which can be used if you are willing to provide your own data. But there is no separate API for trading through a broker--only the browser-based backtester and trading interface is available.
I haven't used it, but I gather that some Quantopian users have applied zipline extensively, which would allow you to customize your development environment. Note:
Discussion of the project is held at the Google Group, [email protected], https://groups.google.com/forum/#!forum/zipline.
Grant