KiCAD Component Pad Clearances
KiCad (version 5) works on a maximum required clearance model.  That means that it will select the smallest clearance possible from the combination of requirements.  In your case, although your pad has small enough clearance, I suspect that your zone fill has a higher clearance set.

In the image below, you'll need to set two things: the clearance and the minimum width.  Because you want the copper to flow between the pads, you'll need the width to be smaller than the space available between pads (when including their clearance.



You said that you don't want this to affect other things on the board.  In that case, you can set the minimum clearance for other elements using the "Design Rules -> Net Class Editor" clearance setting.  Then, as long as your other footprints are set to a clearance of "0" (and their pads are set to "0"), it will use the netclass value.



Viewing the Soldermask

To view the soldermask, you need to have the soldermask layer enabled (and probably selected).  The check-mark enables it but because it is drawn behind the pads, they typically obscure the layer unless you also click on it to make it the front-most layer drawn.