I think the jack is connected properly. Since you hear the pickups (even if they are quiet) it's probably installed correctly. You can probably tell that I don't like barrel jacks. They are literally then only kind of jack that has failed that in one of my basses. My G&L came with one, and I actually modified it to use a Les Paul jack plate and open frame jack. But they work fine for a lot of people.
The DiMarzios may be louder. On the other hand if it's too quiet, you may want to at least be able to adjust the pickup height. In my case it usually takes me a few weeks to decide what height I like the pickups best.
I have only worked on one fretless neck, and it was originally fretted. After I pulled the frets and put some veneer in the fret slots, I had a lot of sanding to do. Once I started, I could see that it was much more uneven than I had imagined. The sanding takes a while, but is not difficult. Mine is not bound, however, and I can see how you might not want to have to re-paint the binding. As to the finish, I think I am a bit odd her in that I did not put finish on my fingerboard. It is rosewood, and very hard. I am using flat-wound strings. If the strings wear the fretboard I figure I can just sand it again. I can say that I played an upright bass for many years with an unfinished rosewood fingerboard without problems. Rosewood is pretty hard ;-)