Wow, you guys are fast.
For finishing, I was thinkig of just using a semigloss urethane so it would look like sandstone. My trials on scrap pieces glued together seem to indicate that the MDF really soaks the urethane up. So, I am now thinking that some sealer would be a good move.
I don't know about pva glue, but what about ordinary cabinet maker's glue, or white glue that dries clear?
The point about the interior of the MDF being softer is quite true. I've had to variously use sandpaper, a rasp/file, and a rotary tool to sculpt as necessary. I don't need it to be absolutely smooth, and some roughness actually fits the look.
By the way, this thing is really an end table - imagine the irregularly shaped 'egg' (or is it a boulder?) but with flat top and bottom surfaces. The egg is irregularly open to the hollow interior on one side, and a shelf is situated halfway up.