I am going to open up a hugh can of worms here as just about every sidecar builder has their own ideas on dimemsional relationships, but please remember that building and setting up a sidecar is more of a black art than a science. My experience comes from having build about a dozen sidecars, the majority of which were trials sidecars, but I have also built a few trail riding outfits and some pseudo MX outfits on 500 Yamahas and currently have a long term Yamaha XS1100 sidecar project gathering dust in the shed.
In regards to widening the sidecar it will make the outfit more stable, but a lot of the "wallowing" that trials outfits do is a result of soft suspension.
The basic relationship of wheel track to wheelbase that I have worked around is that the wheel track should be between 55 to 65% of the wheelbase. The wheelbase of the average observed trials bike from the late 70's - early 80's (the era that I build most of my trials outfits) was around the 1320mm mark. The minimum track for an observed trials sidecar in the GCR's was 750mm (and still is). This minimum wheel track on a 1320mm wheelbase is a relationship of about 57%. I built most of my sidecars right on the minimum wheel track as it made life easier squeezing the outfit between thing. Pushing the wheel track out to 65% of the wheelbase would give a wheel track of around the 850mm mark which would greatly increase the stability of the sidecar when turning right handers. The other dimension that is important in terms of stability when turning to the right is the sidecar wheel lead. The further forward the sidecar wheel the more stable the outfit will be when turning to the right, but you can go to far, and it then give problems when turning left. I usually set the sidecar wheel lead at between 22 to 24% of the wheelbase. Most of the time the final lead was determined by trial and error based on the sidecars ability to turn full lock to the left with out scrubbing the sidecar wheel, which of course leads into the amount of toe-in that the sidecar wheel has, which I usually set at around 2 - 3%.
Sidecar wheel size was always a matter of personal choice. I built most of my trails and trail riding outfits with 16" sidecar wheels, although I used a 14"wheel on my current Bultaco, which I built in 1994. Bigger diameter wheels ride better over bumps, as smaller wheels tend to "fall" into the bumps / holes. Of course, large diameter wheels need bigger / taller mudguards to cover them, so it is always a trade off.
A few people back in my time experimented with sidecar wheel brakes, but setting them up was just way more hassle than they were worth. I rode the final few years of my career on a 350 Bultaco outfits (and still ride a couple of events a year on the same outfit), and brakes that worked would have been a luxury. Back then, as now, I use a decompressor instead of the brakes, which are only there to get the outfit through machine examination. A lot of the modern trials outfits are now running with sidecar wheel brakes, using small disc brake setups.
CJ