Hello,
I notice in the Eagle PCB library, that SMD pads have a TStop layer around then and over them (Solder mask)
Do you know what would happen if this solder mask was omitted from the pads.?
-would the solder then “run away” and short to adjacent pads?
Unless you are using solder paste + screen, I don’t think it matters too much. I’ve ordered boards with and without this layer on the part. Both looked the same.
You’d end up with a board without mask 
There will be some board issues. You’d also likely want to finish the entire board (HASL) or at least spray some OSP on it to prevent the copper from corroding. Doing surface mount reflow will be difficult.
There are some “bare bones” board places which will do mask-less boards for cheap prices.