EAGLE CAD Connecting fills to components and leave traces bare
A copper pour looks like this for example: -



Notice how it skirts around component pads/traces that it is not meant to connect to but connects (via thermals), to the pads it is intended to connect to. My point is this - have you properly set the net name of the pour (or fill if in fact you mean a copper fill) to that net it should connect to.

For the solder mask, you have to create keep-out areas that follow the shape of the pour (or fill) to have exposed copper.