If I ignore DRC errors, can routing go through a copper pour or pad? I'd like large-ish pads at the slot edges for joining boards together at 90 degree angles.
That sounds like it is not really a DRC error in the eyes of the manufacturer. Even in most PCB tools you can turn that off so it is not a error by for example connecting the pour or pad to the same net as the routing so it no longer counts as a 'short'.
If I omit the solder mask over an area on both sides, does the PCB material appear brownish? It's hard to tell from the boards I have with full solder mask.
Do you mean over copper or over bare substrate? Substrate is usually FR4, and I don't think JLC can do anything else (at least not for their cheap 2 dollar boards). It should look like any other FR4, and will be something similar to this:
Source:
http://www.gar-fond.com/english/unclad-fr4.htmlAs for the other two: You should contact the manufacturer on this.