I have done R2R2R from the south twice, both times in April. Don't know about this year but both times I encountered snow on both rims. You gear list mentions a "high traffic" trail so I will assume you will be on the main corridor i.e. North Kaibab Trail.
The first time, 2006, I had snow from the Supai Tunnel all the way to the top. I did not have snowshoes and it was posthole hell. The second time, 2010, the snow was much lighter and I had lightweight snowshoes. The going was sooooo much easier. I was more than happy to schlep them across the G.C. just for that last mile and a half or so to the North Rim.
If you plan to camp on the North Rim, you may very likely be camping on snow. Your Neoair alone may not be up to the job.
The first few miles of the South Rim trails can be very icy in April. Consider some microspikes or similar.
I did not see more than 3-4 people after Cottonwood Campground and I had the North Rim to myself both times.
You are right about temperature swings. The last day on my way out in 2010 it was in the 80's at Phantom Ranch and I topped out on the South Rim in a snowstorm several hours later.