It depends on the pattern whether I leave the papers on while sewing them together. If there are bias edges, YES, it helps keep them from stretching. After I sew them together, I will remove the teeny tiny little triangles. You will see what I mean in the seam lines. They can get buried after sewing the rows together and can be difficult to remove later. The rest are easily removed later. I leave the papers alone to atleast the first border is on. Someimes until the whole top is together.
I would think wetting the paper would make a mess. I use copy paper, I tear along one seam line, then fold it on the next, unfold and tear from an angle like you do paper out of a tablet. Don't be too aggressive, you can distort or loosen stitches.