With regard to the scorched backhoe, oily rags in even a small pile will spontaneously combust; I must have metal cans with self-closing lids in my auto shop for the rags, for this reason. I doubt the bag of concrete affected the process.
Regular concrete that has aged 30 days and has been hosed a few times isn't going to be very alkaline on the surface. Specialty concretes such as high early strength formulas, and shotcrete/gunnite, are much more alkaline when wet, but once they've aged, again, the surface isn't going to be strongly alkaline. The only way it could affect the trailer is if the surface is pulverized from wear and the dust blows up and clings to the trailer- not very likely. The trailer is more likely to be damaged from uncured concrete dust wafting out as bags are poured into the mixer. Rubber isn't very sensitive to moderately strong alkalai, or acid for that matter. However, old-fashioned rubber battery cases, unlike modern plastic ones, were known to allow self-discharging of the battery when left on concrete, hence the (obsolete) admonition to not store batteries there. Maybe that concept got blended into someone's mind, regarding parking the trailer there.
Asphalt was mentioned; you could make a case that hydrocarbons leaching from it could weaken the rubber at the contact point, especially with new asphalt. Some of these hydrocarbons might even condense on the trailer, leaving a kind of waxy or oily film. This would actually protect bare metal from corrosion, but would tend to attract dirt. It could possibly damage clearcoat over time, in a carport-type setting with low ventilation- a bit of a reach.
Some types of tires are more prone to cold "flat spots" than others- polyester versus nylon cords- regardless of the surface parked upon. If the tire is parked in the same spot for months, it could cause localized cracking of the rubber, and a less temporary flat-spot effect, again regardless of the type of surface.
It seems I love the mountains and deserts more than my friends do. I sure miss them!
1971 Streamline Imperial project "Silver Snausage", 1985 Coleman tent trailer, 1964 Little Dipper, 1975 Northwest "Proto Toyhauler", 2004 Harbor Freight folding, still seeking my Airstream.