The jack and lug wrench does not seem to be standard equipment on small trailers. The tow vehicle already has them. Make sure they fit.
The flats do happen while towing, maybe not too often. I know of two, both on the gravel section of the Trans Labrador Highway. One on Norm's Scamp
(if I remember correctly), the other was the right rear on my truck. The pressure sensor saved my bacon, I was able to patch the tire on the truck without even jacking it up, with the patching cord and pump it up with the cheap 12V compressor. Don't leave home without them.
Another two I know of, one was somebody in Florida, who was telling me he had to have the Scamp
shell fixed after his right tire shredded and messed it up. The second was somebody I also talked to in Florida, on another trip, who had to have a damaged axle
and was without a spare at that time, the blown up tire was on the back of his Scamp, showing uneven wear. It is possible that both of those may have been avoided by careful monitoring of the temperature, pressure and wear pattern.