1. Will replacing the battenless, roachless mainsail with a full-batten main with roach likely increase weather helm?
I think it will, but the difference will be quite slight. I think that you'll find the increased lift from the better sail shape will help the boat track more steadily.
2. Are there any good reasons for having no battens and no roach on the mainsail of a cruising boat? None of the reasons I have heard seem very persuasive to me.
You mention that this is a "one off" cruising boat. It is possible that the sails were not made by a commercial loft, in the interests of economy. The only logical reason I can think of for no roach and no battens is to save money.
3. Is a cutter-designed boat, with the mast positioned further aft, inevitably going to have some weather helm when sailed as a sloop.
If the foresail is too small (i.e. - if you are using one of the cutter headsails) - yes. You could probably balance it with a 130 (or larger) genoa.