hmmm, So I think I won't get the solution from WIS...
I think the answer is that it probably depends on the watch and it probably depends on the winder.
Some winders have poorly designed or manufactured motors which expose the watch to magnetism. Some watches are very susceptible to magnetism. A good winder will not expose the watch to any stronger magnetic field than general every day wear. Though I would say that the magnetic field permeating from a small motor like the ones used on a winder would be weaker than the magnetic field from your mobile phone speaker, so holding your phone would expose the watch to a stronger magnetic field. And every modern and not awful winder will have a metal shield between the magnet and the watch anyway, so there is no risk worth worrying about here.
Some winders have limited clearance around the strap or bracelet area and some watches are chunky. This should be easy to spot as an issue before you use it.
In terms of wear and tear, some watches would probably suffer more from being left 'dead' and some more from being constantly 'used', or wound and running.The difference is likely to be negligible. And nobody would ever have done a measured study of a large sample to tell you anything other than a theoretical indication.
If it is a good winder and a good modern watch I wouldn't worry. If it is a good winder and an old watch then I would still use one if setting the complications (date, moonphase etc) were a pain in the arse. I would probably rather just save myself the money and wind it by hand every morning or every other day (depending on the spec) to keep it topped up.
In your case, I would wind your watch by hand each day to keep it going (takes less than 30 seconds) and I would let the Datejust stand idle. If it is only worn every few months then it is a lot of electricity to waste to just keep the date set. I would wind it once a week or every couple of weeks to let it run itself down. I wouldn't worry about the service costs if the oils do suffer because I believe a Rolex service cost is the same if it has stopped working or not - if it is just oil drying or going gunky they will take it apart, clean it, reassemble it and re-oil it anyway, so it shouldn't effect the cost of a service.
If hand winding them is too much of a pain then buy a good winder and don't worry about it.